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e-of-a-hemisphere-of-radius--3x--is-half-the-volume-of-a-full-sphere-58310322026-04-20T21:21:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-angle--120circ--lies-in-the-second-quadrant-using-the-identity--an180circ---heta---an-heta--we-write-1105212026-04-20T21:21:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-first-12-even-numbers-after-2020-are-2022-2024-ldots-2044-the-sum-of-an-arithmetic-sequence-is-s--racn2a1--an-here-n--12-a1--2022-an--2044-thus-s--rac1222022--2044--6-imes-4066--24396-dividing-24396-by-11-11-imes-2216--24376-so-the-remainder-is-24396---24376--20-however-20-mod-11--9-therefore-the-remainder-is-oxed9-65484352026-04-20T21:21:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-prime-factorization-of-2048-is-211-and-1024-is-210-the-lcm-of-two-numbers-is-the-product-of-the-highest-powers-of-all-primes-present-thus-extlcm2048-1024--211--2048-the-answer-is-oxed2048-54503082026-04-20T21:21:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-numbers-congruent-to-3-mod-7-are-3-10-17-ldots-3--7k-the-largest-such-number-leq-100-satisfies-3--7k-leq-100-rightarrow-k-leq-rac977-pprox-13857-thus-k--0-1-ldots-13-giving-14-numbers-the-count-is-oxed14-19855612026-04-20T21:21:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-prime-factorization-of-2025-is-34-imes-52-and-1024-is-210-since-there-are-no-common-prime-factors-the-gcd-is-1-the-answer-is-oxed1-8850942026-04-20T21:21:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sum-of-squares-formula-gives-s--rac15-imes-16-imes-316--1240-compute-1240-mod-13-13-imes-95--1235-so-1240---1235--5-the-remainder-is-oxed5question-a-science-educator-is-designing-a-lesson-on-probability-using-a-digital-simulation-that-generates-random-6-digit-numbers-using-only-the-digits-1-2-and-3-what-is-the-probability-that-a-randomly-generated-number-has-exactly-two-consecutive-digits-that-are-the-same-29047112026-04-20T21:22:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-solve-this-we-must-count-the-number-of-6-digit-numbers-using-only-the-digits-1-2-and-3-such-that-exactly-one-pair-of-consecutive-digits-are-equal-and-all-other-adjacent-digits-are-different-37885912026-04-20T21:21:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-probability-that-the-product-of-three-randomly-chosen-integers-between-0-and-100-is-divisible-by-5-we-use-complementary-probability-first-note-that-an-integer-is-divisible-by-5-if-it-ends-in-0-or-5-among-the-integers-from-0-to-100-there-are-21-multiples-of-5-ie-0-5-10--100-thus-the-probability-that-a-single-randomly-chosen-integer-is-not-divisible-by-5-is-15693482026-04-20T21:35:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-determine-how-many-years-between-1800-and-1900-inclusive-are-divisible-by-4-99660692026-04-20T21:35:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-ways-to-choose-3-papers-from-12-is-9434232026-04-20T21:35:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-ways-to-choose-3-sites-from-10-is-92342312026-04-20T21:35:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-given-sequence-is-an-arithmetic-sequence-with-the-first-term--a--3--and-common-difference--d--5--the-general-term-of-the-sequence-is-given-by-6444862026-04-20T22:27:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-8-and-12-we-first-determine-their-prime-factorizations-73539712026-04-20T22:27:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-smallest-prime-factor-of-77-we-test-divisibility-by-the-smallest-prime-numbers-37403592026-04-20T22:27:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-maximum-value-of-sec-theta--csc-theta2-first-express-it-in-terms-of-sine-and-cosine-3712862026-04-20T22:27:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-68020742026-04-20T22:29:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-point-on-the-line-y--2x--3-closest-to-1--1-we-minimize-the-distance-between-x-2x--3-and-1--1-the-squared-distance-is-6036872026-02-23T10:55:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-y--rac2sin-x--3sin-x--2-solve-for-sin-x-13826532026-04-20T22:54:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-mathbfm--eginbmatrix-a--b--c--d-endbmatrix-45901212026-04-20T22:54:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-expand-25966062026-04-20T22:54:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-given-points-lie-on-the-plane-z--3-the-vector-from-1-2-3-to-1-5-3-is-0-3-0-and-the-vector-to-4-2-3-is-3-0-0-the-fourth-vertex-on-the-same-face-is-obtained-by-adding-these-vectors-1--3-2--3-3--4-5-3-65988152026-04-20T23:03:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product-3x--2-1---14--3x---2---4--3x---6-set-this-equal-to-zero-3x---6--0-implies-x--2-86207352026-04-20T23:03:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-cross-product-mathbfv-times-mathbfa-is-computed-as-25777942026-04-20T23:03:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-equation-sin2x--frac12-has-solutions-when-2x--fracpi6--2pi-n-or-2x--frac5pi6--2pi-n-for-integer-n-solving-for-x-in-0-2pi-21293492026-04-20T23:04:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-dot-product-mathbfu-cdot-mathbfv--mathbfw--mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw-since-mathbfu-mathbfv-and-mathbfw-are-unit-vectors-each-dot-product-is-at-most-1-the-maximum-occurs-when-mathbfu-aligns-with-both-mathbfv-and-mathbfw-ie-mathbfv--mathbfw--mathbfu-then-mathbfu-cdot-mathbfv--mathbfw--1--1--2-21565802026-04-20T23:04:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-5-distinct-parts-a-b-c-d-and-e-and-we-are-to-count-the-number-of-permutations-where-a-appears-before-b-and-c-appears-before-d-83460242026-04-20T23:07:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-8-segment-sequences-using-letters-a-u-c-g-such-that-83340582026-04-20T23:07:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-probability-that-a-single-project-is-exceptionally-strong-ie-gets-h-in-at-least-5-of-the-3-categories-but-there-are-only-3-categories-so-getting-high-in-at-least-5-is-impossible-55027702026-04-20T23:07:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-robot-takes-8-steps-each-independently-choosing-to-go-right-increase-x-or-up-increase-y-uniformly-at-random-assume-fair-walk-the-endpoint-is-x-8---x-for-x--01dots8-so-x-follows-a-binomial-distribution-x-sim-textbinomial8-frac12-4543482026-04-20T23:07:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-number-of-4-protein-combinations-from-6-proteins--binom64--15--61429922026-04-20T23:07:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-distinct-ranking-configurations-of-5-projects-where-48227612026-04-20T23:07:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-smallest-positive-angle-at-which-the-robot-returns-to-its-original-orientation-we-must-find-the-least-common-multiple-lcm-of-the-two-rotation-increments-12circ-and-18circ-first-factor-both-numbers-4144852026-04-20T23:34:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-of-the-city-is-2025-km-and-we-are-to-find-how-many-integer-side-lengths-s-correspond-to-square-grid-sizes-such-that-s2-divides-2025-this-requires-finding-the-number-of-positive-divisors-of-2025-since-each-such-divisor-s2-implies-s-is-an-integer-divisor-of-sqrt2025--45-57065212026-04-20T23:34:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-us-make-the-substitution--x--sqrtu--so-that--u--x2--substituting-into-the-equation-gives-52765662026-04-20T23:35:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-that-12532002026-04-20T23:35:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-a-right-triangle-with-legs--a--and--b--and-hypotenuse--c--the-radius--r--of-the-inscribed-circle-is-given-by-32733072026-04-20T23:48:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diagonal-of-the-square-is-the-diameter-of-the-circle-for-a-square-with-side-length--s--the-diagonal--d--is-60657702026-04-20T23:48:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s--is-43618972026-04-20T23:48:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-area-using-herons-formula-let--a--7---b--10---c--13--the-semi-perimeter--s--is-21003332026-04-20T23:48:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-gears-align-we-need-the-least-common-multiple-lcm-of-their-number-of-teeth-24-and-36-first-factor-both-numbers-90749422026-04-20T23:55:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-find-the-least-common-multiple-lcm-of-6-8-and-12-factor-each-number-47343512026-04-20T23:55:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-geometric-sequence-with-first-term-a--1-and-common-ratio-r--3-the-total-number-infected-by-the-end-of-the-5th-generation-is-the-sum-of-the-first-5-terms-20841702026-04-20T23:55:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-list-the-positive-factors-of-24-1-2-3-4-6-8-12-24-but-digits-must-be-single-digit-numbers-09-so-valid-digits-are-1-2-3-4-6-8-86779102026-04-20T23:55:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-of-the-three-expressions-is-74573262026-04-20T23:55:52+00:00daily0.8https://iqdig1.iqnection.com/resources/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-the-fertilizer-amount-maximizing-yield-is--boxed3--kg-79075522026-02-23T13:50:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--8---s--frac6sqrt8--frac62sqrt2--frac3sqrt2--rationalize--frac3sqrt22--final-answer--boxeddfrac3sqrt22--88402412026-04-21T01:13:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-y-intercept-occurs-when--x--0--substitute--t--frac40--10---2--frac1-2---frac12--the-y-intercept-is--boxedleft0--dfrac12right--28017522026-04-21T01:13:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-minimum-occurs-at--x---fracb2a---frac-2025--2--verify--w2--54---202--45--20---40--45--25--the-smallest--x--is--boxed2-question-a-hydrologist-models-groundwater-flow-with-the-function--fx--sec-x--cos-x2--csc-x---sin-x2--for--0--x--fracpi2--find-the-minimum-value-of--fx--26076032026-04-21T01:13:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-function-15028162026-04-21T01:13:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-function--st--left-sinleftfrac3pi7-tright--cosleftfrac4pi7-tright-right--has-period--t--where-both-components-repeat-the-fundamental-period-is-the-least-common-multiple-of-the-periods-of--sinleftfrac3pi7-tright--and--cosleftfrac4pi7-tright--54075232026-04-21T01:13:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the--y--intercept--b--substitute-the-point--10-2--into-the-equation-60893592026-04-21T01:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-maximum-of--px---x2--4x--m--occurs-at-the-vertex-the--x--coordinate-of-the-vertex-is--x--frac-b2a--frac-4-2--2--substitute--x--2--into--px--34658072026-04-21T01:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-up-equations-using-the-given-points-44571842026-04-21T01:41:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-numerator-and-denominator-by-the-conjugate--sqrt7--sqrt3--25351852026-04-21T01:41:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-de-moivres-theorem--zn--cosleftnthetaright--i-sinleftnthetaright--here--z--cosleftfracpi6right--i-sinleftfracpi6right--so--z6--cosleft6-cdot-fracpi6right--i-sinleft6-cdot-fracpi6right--cospi--i-sinpi--evaluating--cospi---1--and--sinpi--0--so--z6---1--0i--2752232026-04-21T02:07:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set--frac18d0--d0-cdot-2-t12--divide-both-sides-by--d0---frac18--2-t12--note--frac18--2-3--so--2-3--2-t12--equate-exponents---3---fract12--solve-for--t---t--36--71295222026-04-21T02:07:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-closest-point-occurs-where-the-vector-from--5-7--to-the-line-is-perpendicular-to-the-direction-vector--beginpmatrix-3--4-endpmatrix--let-the-point-on-the-line-be--2--3t--1--4t--the-vector-from--5-7--to-this-point-is--beginpmatrix-2--3t---5---1--4t---7-endpmatrix--beginpmatrix--3--3t---8--4t-endpmatrix--dot-product-with--beginpmatrix-3--4-endpmatrix--must-be-zero--3-3--3t--4-8--4t--0--simplify---9--9t---32--16t--0-rightarrow-25t---41--0-rightarrow-t--frac4125--substitute-back--x--2--3-cdot-frac4125--frac17325---y---1--4-cdot-frac4125--frac15625--62501762026-02-23T14:50:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero--x-cdot-3--2-cdot--1--0--simplify--3x---2--0-rightarrow-x--frac23--40741872026-04-21T02:07:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-by-solving--2cos2theta--1--0-rightarrow-cos2theta---frac12--the-general-solutions-for--2theta--are--2theta--120circ--360circ-k--or--2theta--240circ--360circ-k--where--k-in-mathbbz--dividing-by-2--theta--60circ--180circ-k--or--theta--120circ--180circ-k--for--theta-in-0circ-360circ--the-solutions-are--60circ-120circ-240circ-300circ--10729132026-04-21T02:07:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-rewrite--fx--fracsin2-x--4sin-x--sin-x--frac4sin-x--let--y--sin-x-in-0-1--the-function-becomes--fy--y--frac4y--the-derivative--fy--1---frac4y2--has-critical-point-at--y--2--but--y-leq-1--analyze-endpoints-as--y-to-0---fy-to-infty--at--y--1---f1--1--4--5--the-minimum-is--5--98016382026-04-21T02:07:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--vecoa--mathbfa---vecob--mathbfb--then--overrightarrowoc--mmathbfa--nmathbfb--for--overrightarrowoc-perp-mathbfa---mathbfb--their-dot-product-is-zero--mmathbfa--nmathbfb-cdot-mathbfa---mathbfb--0--expand--mmathbfa2---mmathbfa-cdot-mathbfb--nmathbfb-cdot-mathbfa---nmathbfb2--0--substitute--mathbfa--2---mathbfb--3---mathbfa-cdot-mathbfb--2-cdot-3-cdot-cos-60circ--3---4m---3m--3n---9n--0-rightarrow-m---6n--0--choose--n--1--then--m--6--normalize-if-needed-but-the-relation-is--m--6n--for-simplicity-take--n--1---m--6--37815312026-04-24T12:38:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-measurement-be--u--a-positive-multiple-of-5-such-that--u3--1500--85153452026-04-24T12:38:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-of--3u2---5u7--and--4u1--is-given-by-26104512026-04-24T12:38:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-mathbfv--langle-v1-v2-v3-rangle-compute-mathbfv-times-mathbfb-58193852026-04-21T02:10:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-magnitude-of-the-cross-product-is-given-by-mathbfu-times-mathbfw--mathbfu-mathbfw-sin-theta-here-theta--60circ-99373642026-04-21T02:10:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-a--b--2025-the-greatest-common-divisor-gcda-b-must-divide-a--b--2025-the-maximum-possible-value-of-gcda-b-occurs-when-a-and-b-are-both-multiples-of-the-largest-divisor-of-2025-since-2025--452--34-cdot-52-its-largest-proper-divisor-is-2025--3--675-thus-gcda-b--675-when-a--675-and-b--1350-31510212026-04-21T02:19:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sum-of-cubes-formula-is-leftfracnn12right2-for-n--12-the-sum-is-leftfrac12-cdot-132right2--782--6084-dividing-by-13-6084-div-13--468-with-remainder-0-since-13-cdot-468--6084-59732122026-04-21T02:19:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-least-common-multiple-lcm-of-14-and-21-is-42-the-smallest-three-digit-multiple-of-42-is-42-times-3--126-41035422026-04-21T02:19:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-prime-factorization-16--24-24--23-cdot-3-lcm-takes-the-highest-powers-24-cdot-3--16-cdot-3--48-24578382026-04-21T02:19:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-prime-factorization-108--22-cdot-33-144--24-cdot-32-gcd-takes-the-lowest-powers-22-cdot-32--4-cdot-9--36-30899172026-04-21T02:19:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-and-arranging-instruments-for-display-first-choose-3-telescopes-from-8-inom83-then-choose-2-microscopes-from-5-inom52-after-selection-arrange-the-5-chosen-instruments-in-a-row-5-ways-45746962026-04-21T02:27:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-number-of-ways-to-choose-5-varieties-from-10-inom105-91826722026-04-21T02:27:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-kiosks-are-arranged-in-a-circle-and-rotations-are-considered-the-same-we-fix-one-session-to-eliminate-rotational-symmetry-first-assign-4-out-of-7-kiosks-to-the-sessions-inom74-cdot-4-would-count-linear-arrangements-but-due-to-circular-symmetry-we-divide-by-4-the-number-of-rotations-12004002026-04-21T02:27:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-find-the-constant--k--using-the-given-values-34633072026-04-21T02:27:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-expression--cos-x--isin-x--corresponds-to--eix--on-the-unit-circle-in-the-complex-plane-therefore--cos-x--isin-x--1--eix--1--the-modulus-is--eix--1--using-the-identity--eix--1--sqrtcos-x--12--sin2-x--we-simplify-93132582026-04-24T12:38:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since--sin2-x--cos2-x--1--we-write--fx--frac1sin2-x--4cos2-x--frac11--3cos2-x--let--y--cos2-x--so--0-leq-y-leq-1--then--fx--frac11--3y--as--y--varies-from-0-to-1--1--3y--varies-from-1-to-4-so--fx--varies-from--frac14--to--1--hence-the-range-of--fx--is--leftfrac14-1right--the-final-answer-is--boxedleftfrac14-1right--98878732026-04-22T06:04:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-with--tan-z--cot-z--2--recall--tan-z--fracsin-zcos-z---cot-z--fraccos-zsin-z--then-59618922026-04-22T06:04:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-lets-simplify-each-term-first--tan-x--cot-x--fracsin-xcos-x--frac-5808362026-04-21T02:29:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-given-points-lie-on-the-base-face-of-the-cube-in-the-xy-plane-the-fourth-vertex-of-the-square-face-must-complete-the-rectangle-the-missing-vertex-is-110-as-it-forms-equal-sides-and-right-angles-with-the-given-points-38904052026-04-21T02:34:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-solve-2x--60circ-120circ-420circ-480circ-within-0circ-leq-2x-leq-720circ-thus-x--30circ-60circ-210circ-240circ-sum-30--60--210--240--540circ-12417082026-04-21T02:34:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-scalar-triple-product-equals-the-volume-of-the-parallelepiped-maximum-occurs-when-vectors-are-orthogonal-giving-mathbfa-cdot-mathbfb-times-mathbfc--1-cdot-1-cdot-1-cdot-sin90circcos0circ--1-835842026-04-21T02:34:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-problem-requires-partitioning-7-distinguishable-solutions-into-3-indistinguishable-non-empty-vials-this-is-equivalent-to-the-stirling-numbers-of-the-second-kind--s7-3--the-formula-for-stirling-numbers-is-23161602026-04-21T02:34:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-modules-are-indistinguishable-and-empty-modules-are-allowed-the-number-of-ways-is-the-sum-of-stirling-num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e-has-5-positions------the-middle-3rd-character-is-fixed-as-b-we-must-choose-4-distinct-letters-from-a-b-c-d-e-with-b-fixed-in-position-3-so-the-remaining-4-positions-use-the-other-4-letters-but-b-is-already-placed-so-we-use-a-c-d-e-for-the-other-4-positions-44815682026-04-21T11:31:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-9-tokens-in-total-where-there-are-4-identical-gold-3-identical-silver-and-2-identical-bronze-tokens-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-15071442026-04-21T11:31:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-7-badges-where-3-are-identical-robotics-r-2-are-identical-coding-c-and-2-are-identical-data-science-d-the-number-of-distinct-sequences-is-the-multinomial-coefficient-90436792026-04-21T11:31:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-given-equation-in-spherical-coordinates-is--22302742026-04-21T11:44:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-vector-equation-mathbfv-imes-mathbfa--mathbfb-has-a-solution-only-if-mathbfa-cdot-mathbfb--0-since-the-cross-product-is-perpendicular-to-mathbfa-99804622026-04-21T11:44:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-identity-sin2z--2sin-z-cos-z-so-47078242026-04-21T11:44:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set--3t--9t-1--since--9--32--rewrite-the-equation-as--3t--32t-1--simplify-the-right-hand-side--3t--32t-1--equate-the-exponents--t--2t---1--solve--t--2t---2--so--t--2--verify-by-substituting-back--32--91----9--9--thus-the-time-is--boxed2--37736902026-04-21T11:49:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x--be-liters-of-15-solution-and--y--be-liters-of-5-solution-the-system-is-71554122026-04-21T11:49:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set--20--80e-01t--40--subtract-20--80e-01t--20--divide-by-80--e-01t--025--take-natural-logarithm---01t--ln025--solve--t---fracln02501--fracln401-approx-frac138601--1386--round-to-two-decimal-places--boxed1386--minutesthe-analytical-engine-designed-by-charles-babbage-is-considered-a-precursor-to-modern-computers-if-babbages-design-allows-a-program-to-execute-120-distinct-operations-and-each-operation-requires-3-memory-accesses-how-many-total-memory-accesses-are-needed-to-complete-the-program-once-13607172026-04-21T11:49:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-group-and-complete-the-square-98097562026-04-21T11:50:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use--z2--w2--z--w2---2extrez-overlinew--compute--z--w2--2--4i2--4--16--20--let--z-overlinew--a--bi--then--extrez-overlinew--a--from--z--w--2--4i--and--zw--13---2i--note--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--2--4i2---2a--20---2a--also--zw--overlinezw--2extrezw--26--but-this-path-is-complex-alternatively-solve-for--z2--w2--z--w2---2extrez-overlinew--however-using--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--z--w2---2extrez-overlinew--since--z-overlinew--overlinez-w--2extrez-overlinew--and--z--woverlinez--overlinew--z2--w2--z-overlinew--overlinez-w--z2--w2--2extrez-overlinew--let--s--z2--w2--then--20--s--2extrez-overlinew--from--zw--13---2i--take-modulus-squared--zw2--169--4--173--z2-w2--let--z2--a---w2--b--then--a--b--s---ab--173--also--s--20---2extrez-overlinew--this-system-is-complex-instead-assume--z--and--w--are-roots-of--x2---2--4ix--13---2i--0--compute-discriminant--d--2--4i2---413---2i--4--16i---16---52--8i---64--24i--this-is-messy-alternatively-use--z2--w2--z--w2--z---w2---2z-overlinew--but-no-correct-approach--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--20---2extrez-overlinew--from--z--w--2--4i---zw--13---2i--compute--z-overlinew--overlinez-w--2extrez-overlinew--but--z--woverlinez--overlinew--20--z2--w2--z-overlinew--overlinez-w--s--2extrez-overlinew--let--s--z2--w2---t--extrez-overlinew--then--s--2t--20--also--z-overlinew--zw--from--zw--sqrt173--but--t--extrez-overlinew--however-without-more-info-this-is-incomplete-re-evaluate-use--z2--w2--z--w2---2extrez-overlinew--and--extrez-overlinew--extreraczww-overlinew-cdot-overlinew2--too-complex-instead-assume--z--and--w--are-conjugates-but--z--w--2--4i--implies--z--a--bi---w--a---bi--then--2a--2-rightarrow-a--1---2b--4i-rightarrow-b--2--but--zw--a2--b2--1--4--5-12101832026-02-24T01:32:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-rationalize-the-denominator-multiply-numerator-and-denominator-by-the-conjugate-sqrt7--sqrt2-22265622026-04-21T12:04:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-with-the-identity-34919382026-04-21T12:04:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-distributive-property-50712752026-04-21T12:04:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--ht--t2---4t--5m--and--h3--10--substitute--t--3--into-the-expression-21656952026-04-21T12:04:55+00:00daily0.8http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arrangements-for-the-entities-the-letter-pair-and-3-manuscripts-the-letters-within-the-pair-can-be-ordered-in-2-ways-total-favorable-arrangements-4-imes-2-total-possible-arrangements-5-the-probability-is-rac4-imes-25--rac25-oxeddfrac25-96049642026-04-21T14:55:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-there-are-11-integers-010-of-which-6-are-even-0-2-4-6-8-10-the-number-of-ways-to-choose-4-distinct-even-integers-is-inom64-the-total-number-of-ways-to-choose-4-distinct-integers-is-inom114-the-probability-is-racinom64inom114--rac15330--rac122-oxeddfrac122-16089312026-04-21T14:55:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-letters-26-vowels-5-consonants-21-total-3-letter-arrangements-with-distinct-letters-26-imes-25-imes-24-unfavorable-cases-no-vowels-21-imes-20-imes-19-favorable-probability-1---rac21-imes-20-imes-1926-imes-25-imes-24--1---rac798015600--1---rac133260--rac127260-oxeddfrac127260-67831312026-04-21T14:55:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-number-of-ways-to-choose-3-18th-century-manuscripts-is-inom53-total-ways-to-choose-3-manuscripts-from-15-is-inom153-the-probability-is-racinom53inom153--rac10455--rac291-oxeddfrac291question-if-the-combined-population-of-two-marine-mammal-species-in-a-region-is-10-and-the-sum-of-their-squared-populations-is-50-find-the-sum-of-their-cubes-32957642026-04-21T14:55:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-populations-be--a--and--b--given--a--b--10--and--a2--b2--50--first-compute--ab--using--a--b2--a2--2ab--b2--substituting--100--50--2ab--so--ab--25--the-sum-of-cubes-is--a3--b3--a--b3---3aba--b--1000---3-cdot-25-cdot-10--1000---750--oxed250--75534412026-04-21T14:55:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given--x--y--45--multiplying-by-5-yields--5x--5y--5x--y--5-cdot-45--oxed225--62436542026-04-21T14:55:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-numerator-and-denominator-by--sqrt7--sqrt3--64861212026-04-21T14:55:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-recognize-it-as-a-difference-of-cubes--2a3---3b3--8a3---27b3--alternatively-expand-directly-51265052026-04-21T14:55:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set--3x--2---2x--12--solve--5x--10--so--x--2--substitute-into--y--32--2--8--the-intersection-is--oxed2-8-question-let--gx--be-a-polynomial-such-that--g2x--3--4x2--12x--5--find--gx2---1--32095712026-02-24T04:43:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--u--2x--3--then--x--racu---32--substitute-into--gu--4leftracu---32-42285332026-04-21T14:55:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-group-terms--9x2---2x---16y2--4y--144--complete-the-square--9x---12---1---16y--22---4--144--expand--9x---12---9---16y--22--64--144--simplify--9x---12---16y--22--89--the-center-is-at--1--2--final-answer-oxed-50422852026-02-24T04:43:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-number-must-be-divisible-by-the-least-common-multiple-lcm-of-7-11-and-13-since-these-are-all-primes-lcm--7--11--13--1001-the-smallest-four-digit-number-is-the-first-multiple-of-1001--1000-which-is-1001-imes-1--1001-thus-the-answer-is-oxed1001-28726502026-04-21T14:57:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-greatest-common-divisor-of-a-and-b-must-divide-their-sum-a--b--100-the-largest-divisor-of-100-is-50-to-achieve-gcda-b--50-set-a--50-and-b--50-thus-the-maximum-value-is-oxed50-55117222026-04-21T14:57:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-amount-of-saline-in-the-first-solution-be--020-times-3--06--liters-the-second-solution-contributes--050x--liters-of-saline-the-total-volume-is--3--x--liters-and-the-total-saline-is--0353--x--setting-up-the-equation-71019972026-04-21T15:06:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-model-states-that-each-hour-the-count-is-multiplied-by-3-after-1-hour--31--after-2-hours--32--so-after-4-hours--34--but-the-problem-states-the-population-is--36--times-initial-which-contradicts-unless-growth-rate-differs-however-assuming-the-model-is-correct-after-4-hours-72517452026-04-21T15:06:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--t--6--into-the-function-93473482026-04-21T15:06:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-local-extrema-of-the-function--pt--t3---6t2--9t--10--we-first-compute-its-derivative-29001922026-04-21T15:06:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-slant-asymptote-of--it--fract2--4t--2--we-perform-polynomial-long-division-of--t2--4--by--t--2--70588662026-04-21T15:07:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-maximum-of--ft--frac100tt2--4--we-compute-its-derivative-using-the-quotient-rule-23477552026-04-21T15:07:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-analyze-the-cubic-polynomial--rx--x3---3x--2--to-find-the-number-of-real-roots-we-first-attempt-rational-root-theorem-possible-rational-roots-are--pm1-pm2--testing-1183632026-04-21T15:07:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--iu--u---fracu39--and--ixn--frac1n--so-81624422026-04-21T15:07:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-standard-form-of-a-parabola-with-vertical-axis-is--y--frac14px2--the-focus-is-at--0-p--so--p--5--the-width-of-20-units-implies-the-parabola-passes-through--10-5--substituting--5--frac120100-rightarrow-5--5--which-holds-thus-the-equation-is--y--frac120x2--boxedy--dfrac120x2-63613262026-04-21T15:12:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-partial-fractions--frac1kk2--frac12left-frac1k---frac1k2-right--the-sum-becomes--frac12-left-sumk150-frac1k---sumk150-frac1k2-right--frac12-left-sumk150-frac1k---sumk352-frac1k-right--telescoping-gives--frac12-left-frac11--frac12---frac151---frac152-right--frac12-left-frac32---frac1032652-right--frac12-cdot-frac39752652--frac13251768--boxeddfrac13251768-82469492026-04-21T15:12:15+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-identity--p2--q2--p--q2---2pq--substituting--10---24i--3--4i2---2pq--compute--3--4i2--9--24i---16---7--24i--then--10---24i---7--24i---2pq--rearranging--17---48i---2pq-rightarrow-pq--frac-17--48i2--boxed-dfrac172--24i-94747762026-02-24T04:59:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-probability-follows-a-binomial-distribution-with-parameters-n--4-trials-and-p--frac13-success-probability-the-formula-for-exactly-k-successes-is-binomnk-pk-1-pn-k-for-k--2-58829992026-04-21T15:15:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinct-elements-into-2-non-empty-subsets-which-is-the-stirling-number-of-the-second-kind-s6-2-the-formula-for-sn-2-is-2n-1---1-for-n--6-52692032026-04-21T15:15:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-binomial-distribution-with-n--3-p--frac14-and-k--1-49192562026-04-21T15:15:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-stars-and-bars-problem-the-formula-for-distributing-n-identical-items-into-k-distinct-boxes-is-binomn--k---1k---1-here-n--4-k--3-80352442026-04-21T15:15:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-numerator-and-denominator-by-the-conjugate-of-the-denominator-sqrt7---3-88552932026-04-21T15:15:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-terms-are-a-a--d-a--2d-a--3d-a--4d-16965872026-04-21T15:16:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-equation--x--y--4--describes-a-diamond-or-rhombus-centered-at-the-origin-to-find-the-vertices-consider-the-four-cases-for-absolute-values-22495412026-04-21T15:23:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-standard-form-of-the-ellipse-centered-at-the-origin-is-fracx2a2--fracy2b2--1-where-a--b-given-the-major-axis-is-10-we-have-2a--10-rightarrow-a--5-the-minor-axis-is-6-so-2b--6-rightarrow-b--3-the-distance-from-the-center-to-a-focus-is-given-by-c--sqrta2---b2-substituting-a--5-and-b--3-we-get-c--sqrt25---9--sqrt16--4-19872512026-04-21T15:53:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-maximize--ct--frac3tt2--4--we-take-its-derivative-using-the-quotient-rule-82476892026-04-21T15:53:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-simplify-the-function-61332992026-04-21T15:53:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-angle-between-two-unit-vectors--vecv--and--vecw--is-given-by-the-cosine-of-the-difference-of-their-angles-88158542026-04-21T15:53:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-of-the-parallelogram-formed-by-vectors--veca--and--vecb--is-the-magnitude-of-their-cross-product-47253032026-04-21T15:53:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--x--y--12--and--x2--y2--80--44159012026-04-21T15:57:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--a---b--6--and--a2--b2--130--67242772026-04-21T15:57:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given--m--n--10---m2--n2--58--84085122026-04-24T12:42:24+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-lattice-points-on-the-hyperbola-fracx2144---fracy225--1-multiply-both-sides-by-144-cdot-25--3600-to-eliminate-denominators-40450792026-04-21T15:58:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-mathrmlcm18-24-36-91978152026-04-21T15:58:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-perimeter-of-a-rectangle-is-given-by--p--2l--w--where--l--is-length-and--w--is-width-given--2l--w--80--we-simplify-to--l--w--40--the-area--a--lw--express--w--40---l--so--a--l40---l--40l---l2--this-quadratic-equation-opens-downward-and-its-maximum-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ply-mathbfm-with-beginpmatrix-1--2--3--4-endpmatrix-to-get-18144792026-04-21T22:42:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-pythagorean-theorem-since-overrightarrowoa-and-overrightarrowob-are-perpendicular-compute-overrightarrowoc2--2overrightarrowoa2--overrightarrowob2--425--144--100--144--244-thus-overrightarrowoc--sqrt244--2sqrt61-final-answer-boxed2sqrt61-31654232026-04-21T22:43:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-up-the-system-81442062026-04-21T22:43:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-us-define-a-function--fu--such-that-66248062026-04-21T22:54:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-that--gx--is-a-cubic-polynomial-33099912026-04-21T23:22:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-functional-equation-is-32490262026-04-21T23:22:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since--px--is-a-cubic-polynomial-when-dividing-by--x4---1--the-remainder--rx--must-be-a-polynomial-of-degree-less-than-4-however-because--x4---1--has-roots--pm-1-pm-i--and--px--is-real-the-remainder-can-be-expressed-as-any-cubic-or-lower-but-here-we-are-to-find-the-actual-remainder-which-is-unique-and-of-degree---4--but-since--px--is-already-degree-3-and-division-by-degree-4-polynomial-yields-a-unique-remainder-of-degree---4--and-since--px--agrees-with-any-degree-3-polynomial-at-these-4-points-we-conclude-27516082026-04-21T23:22:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-apply-the-cauchy-schwarz-inequality-in-the-engel-form-also-known-as-titus-lemma-92365532026-04-21T23:22:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-value-of--k--substitute--x--3--into-the-function-and-set--f3--0--31200942026-04-22T00:16:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-population-stabilizes-at--t---6--this-means--p-6--0--substituting-into-the-equation-9708362026-04-22T00:16:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-rationalize-the-denominator-of--frac3x--4sqrtx---2--multiply-numerator-and-denominator-by-the-conjugate--sqrtx--2--84780102026-04-22T00:16:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-expression--x2--5x--6--is-a-quadratic-trinomial-we-look-for-two-numbers-that-multiply-to-6-and-add-to-5-these-numbers-are-2-and-3-71442402026-04-22T00:16:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-a-point--3-11--lies-on-the-line--y--2x--b--substitute--x--3--and--y--11--6892072026-04-22T00:16:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-vertex-of-a-parabola--hx--x2---4x--c--occurs-at--x--frac--421--2--which-matches-the-given-condition-now-substitute--x--2--and-set--h2--3--41638222026-04-22T00:16:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--fx--ax3--bx2--cx--d--we-use-the-given-values-to-form-a-system-of-equations-56292572026-04-22T00:16:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-that--a2--b2--1--and--ab--frac12--we-are-to-find--a4--b4--72758012026-04-22T00:41:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-minimize-53105032026-04-22T00:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--sqrtu--t--so--u--t2--substituting-into-the-equation-80789712026-04-22T00:41:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-in-spherical-coordinates-phi-is-the-polar-angle-measured-from-the-positive-z-axis-when-phi-is-constant-the-set-of-points-forms-a-cone-with-its-apex-at-the-origin-opening-symmetrically-around-the-z-axis-for-phi--fracpi4-the-cone-makes-a-45circ-angle-with-the-z-axis-thus-the-shape-is-a-cone-26017912026-04-22T00:42:24+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-theta-be-the-angle-between-any-pair-of-vectors-since-mathbfu-cdot-mathbfv--costheta--frac12-theta--60circ-the-magnitude-squared-is-mathbfu--mathbfv--mathbfw2--3--2mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw--mathbfv-cdot-mathbfw--3--23-cdot-frac12--3--3--6-thus-the-maximum-value-is-sqrt6-46992202026-04-22T00:42:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-orthogonality-their-dot-product-must-be-zero-1-cdot-x--x-cdot-3--2-cdot--1--0-simplifying-x--3x---2--0-rightarrow-4x--2-rightarrow-x--frac12-94311222026-04-22T00:42:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-angle-aob--90circ-overrightarrowoa-cdot-overrightarrowob--0-compute-overrightarrowoc2--4overrightarrowoa2--overrightarrowob2---4overrightarrowoa-cdot-overrightarrowob--44--9---0--25-thus-overrightarrowoc--5-23456122026-04-22T00:42:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-t--sin-x--csc-x-note-tan-x--cot-x--fracsin-xcos-x--fraccos-xsin-x--frac1sin-x-cos-x-let-y--sin-x-cos-x--frac12sin-2x-so-tan-x--cot-x--frac2sin-2x-the-expression-becomes-leftfrac2sin-2xright2--sin-x--csc-x2-let-z--sin-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ine-of--360circ--corresponds-to-the-x-coordinate-of-the-point-on-the-unit-circle-at-a-full-rotation-from-the-positive-x-axis-this-point-is--1-0--so--cos-360circ--1--99434522026-04-22T01:06:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-of-the-cylinder-is--vtextcyl--pi-r2-6r--6pi-r3--the-volume-of-the-cone-is--vtextcone--frac13-pi-2r2-3r--frac13-pi-4r23r--4pi-r3--the-ratio-is-10101872026-04-22T01:06:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-herons-formula-the-semi-perimeter-is-89980952026-04-22T01:06:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-surface-area-of-a-cone-is-the-sum-of-the-base-area-and-the-lateral-surface-area-the-base-area-is-14996912026-04-22T01:06:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-tangent-of--45circ--is-defined-as--fracsin-45circcos-45circ--fracfracsqrt22fracsqrt22--1--74232022026-04-22T01:06:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diameter-of-the-inscribed-circle-is-equal-to-the-side-length-of-the-square-which-is--8--textcm--therefore-the-radius-is--frac82--4--textcm--the-area-of-the-circle-is-23105242026-04-22T01:06:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-side-length-of-the-hexagon-is--frac366--6--text-44463582026-04-22T01:06:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-of-a-sphere-is-given-by--v--frac43-pi-r3--differentiating-with-respect-to-time--t--we-get-38321262026-04-22T01:12:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-triangle-has-side-lengths--a--13---b--14--and--c--15--first-compute-the-semi-perimeter--s--91412542026-04-22T01:12:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-original-side-length-of-the-equilateral-triangle-is--frac363--12--cm-the-area-of-an-equilateral-triangle-is-given-by-66735942026-04-22T01:12:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-hypotenuse-be--z--25--m-and-the-inradius-be--c--5--m-for-a-right-triangle-the-area--a--is-given-by-89531582026-04-22T01:12:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--3--into--fx--74614992026-04-22T01:32:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--5--into--hx--43155132026-04-22T01:32:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--y--x2---3x--then-the-product-becomes--y--2my--5m--y2--7my--10m2--substitute-back-3938422026-04-22T01:32:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--m---3--48896882026-04-22T01:32:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x--be-the-liters-of-15-fertilizer-and--y--be-the-liters-of-40-fertilizer-the-system-is-94438472026-04-22T01:32:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-width-be--5k--and-height--3k--by-the-pythagorean-theorem-29381982026-04-22T01:32:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-probability-of-neither-test-being-positive-is--left1---rac23-83346262026-04-22T01:32:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-with-a1--1-compute-a2--ma1--1---rac122--rac12-next-compute-a3--ma2--rac12---racrac1222--rac12---rac18--rac38-thus-the-value-of-a3-is-oxeddfrac38-54902422026-04-22T01:32:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-cauchy-schwarz-inequality-22--32--42x2--y2--z2-geq-2x--3y--4z2-this-gives-29x2--y2--z2-geq-144-so-x2--y2--z2-geq-rac14429-equality-holds-when-racx2--racy3--racz4--k-leading-to-x--2k-y--3k-z--4k-substituting-into-2x--3y--4z--12-gives-4k--9k--16k--29k--12-so-k--rac1229-thus-the-minimum-value-is-oxeddfrac14429-17948882026-04-22T01:32:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-a-recursive-sequence-defined-by--bn1--gbn--where--gt--t---ract24--and--b1--1--we-are-to-find--limn-o-infty-bn--assuming-the-limit-exists-8723092026-04-22T01:39:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-a-cubic-polynomial--hx--x3--px--q--and-values-47100382026-04-22T01:39:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-equation-62753782026-04-22T01:39:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--a--2x---b--y--so-the-equation-becomes-63100702026-04-22T01:39:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-angles-are-30circ-60circ-and-90circ-using-the-law-of-sines-frac5sin-30circ--fraccsin-90circ-since-sin-30circ--05-and-sin-90circ--1-c--frac505--10-km-85842342026-04-22T01:44:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-circumference-c--2pi-r-so-r--fracc2pi--frac12pi2pi--6-nm-84957532026-04-22T01:44:05+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-longest-chord-in-a-circle-is-the-diameter-however-the-question-specifies-the-longest-chord-that-is-not-a-diameter-which-would-approach-the-diameter-for-a-circle-the-maximum-chord-length-is-the-diameter-so-the-answer-is-14-cm-18434672026-04-22T01:44:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-hypotenuse-is-sqrt92--122--15-cm-sin-theta--fractextoppositetexthypotenuse--frac915--frac35-54284032026-04-22T01:44:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-largest-number-of-identical-grants-compute-gcd2025-1515-66743352026-04-22T01:44:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-smallest-number-of-distinct-sequences-is-the-least-common-multiple-lcm-of-13-and-17-56100832026-04-22T01:44:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-alignment-occurs-at-the-least-common-multiple-of-2025-and-1515-40768982026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-x2-equiv-25-pmod100-85626942026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-problem-requires-calculating-the-number-of-distinct-permutations-of-10-items-where-there-are-5-identical-bees-3-identical-wasps-and-2-identical-beetles-the-formula-for-multinomial-coefficients-applies-94830882026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-multinomial-permutation-problem-the-total-sequences-are-given-by-32038132026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-total-number-of-ways-to-choose-3-species-binom203--1140-the-number-of-favorable-outcomes-is-binom102-cdot-binom61--45-cdot-6--270-the-probability-is-17599502026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-treat-each-discipline-as-a-block-there-are-3-ways-to-arrange-the-blocks-around-the-table-within-the-biology-block-3-physics-2-and-chemistry-2-for-circular-permutations-fix-one-block-to-eliminate-rotational-symmetry-9956392026-04-22T01:44:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-ways-to-choose-4-species-binom134--715-favorable-outcomes-binom82-cdot-binom52--28-cdot-10--280-the-probability-is-649522026-04-22T01:44:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-treat-the-two-consecutive-sessions-as-a-single-entity-there-are-4-ways-to-arrange-the-4-entities-the-pair-and-the-other-3-sessions-the-pair-can-be-ordered-in-2-ways-thus-total-arrangements-are-4-times-2--24-times-2--48-boxed48-91790412026-04-22T01:54:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-combinations-without-restrictions-dbinom94--126-subtract-combinations-where-both-problematic-startups-are-included-dbinom72--21-choosing-2-more-from-the-remaining-7-valid-combinations-126---21--105-boxed105-75705172026-04-22T01:54:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-inclusion-exclusion-total-assignments-36--729-subtract-assignments-missing-at-least-one-category-3-times-26--192-add-back-assignments-missing-two-categories-3-times-16--3-valid-assignments-729---192--3--540-boxed540question-an-electrical-engineer-tests-a-new-circuit-with-5-identical-resistors-each-with-a-10-probability-of-failing-independently-what-is-the-probability-that-exactly-2-resistors-fail-55642412026-04-22T01:54:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-probability-is-calculated-using-the-binomial-distribution-formula-6834292026-04-22T01:54:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-complementary-probability-of-no-scrolls-containing-the-symbol-is-0958-thus-37768672026-04-22T01:54:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-binomial-formula-45012002026-04-22T01:54:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-first-5-terms-are--a1--5---a2--8---a3--11---a4--14--and--a5--17--their-sum-is--5--8--11--14--17--55--dividing-55-by-7-gives-a-quotient-of-7-and-a-remainder-of--55---7-times-7--55---49--6--68024932026-04-22T01:54:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x-equiv-1-pmod9--and--x-equiv-1-pmod11--this-implies--x-equiv-1-pmod99--since-9-and-11-are-coprime-the-two-digit-numbers-satisfying-this-are--99--1--100--not-two-digit-and--1--however-the-next-smaller-solution-is--1--which-is-not-two-digit-thus-no-two-digit-number-satisfies-both-conditions-12143972026-04-22T01:54:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-first-4-primes-after-20-are-23-29-31-and-37-their-sum-is--23--29--31--37--120--dividing-120-by-5-gives-a-remainder-of--120---5-times-24--0--33620952026-04-22T01:54:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-area-of-a-triangle-with-known-side-lengths--a--7---b--10--and--c--13--we-use-herons-formula-27808442026-04-22T01:54:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--a--b--c--1--with--a-b-c--0--so--1---a--b--c---1---b--a--c--and--1---c--a--b--the-expression-becomes-2113822026-04-22T02:36:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--fx--ax2--bx--c--use-the-given-conditions-92165682026-04-22T02:37:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--u--sqrtv--so--v--u2--and--u-geq-0--substitute-12343512026-04-22T02:37:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-slope-of-the-line-from-2-3-to-5-11-is-frac11---35---2--frac83-since-tan-theta-equals-the-slope-tan-theta--frac83-boxeddfrac83-43336312026-04-22T02:37:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-solve-2z--60circ-120circ-420circ-480circ-within-0circ-720circ-yielding--11137272026-04-22T02:37:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-smallest-positive-integer-divisible-by-both-7-and-5-we-compute-the-least-common-multiple-lcm-of-7-and-5-since-7-and-5-are-coprime-their-lcm-is-simply-their-product-74103032026-04-22T02:43:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-that-the-total-number-of-regions-is-12-and-the-number-of-clusters-must-be-an-integer-between-2-and-5-inclusive-we-want-to-maximize-the-number-of-regions-per-cluster-which-means-minimizing-the-number-of-clusters-while-satisfying-the-constraints-23565392026-04-22T02:43:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-180-dots-and-must-arrange-them-into-rows-with-the-same-number-of-dots-where-the-number-of-rows-is-a-prime-number-greater-than-3-to-maximize-the-number-of-dots-per-row-we-must-minimize-the-number-of-rows-among-valid-prime-numbers-69479582026-04-22T02:43:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-divide-both-48-and-72-regions-into-equal-sized-groups-with-no-regions-left-out-we-must-find-the-greatest-common-divisor-gcd-of-48-and-72-57959782026-04-22T02:43:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-smallest-number-of-samples-per-column-such-that-the-number-of-columns-is-a-multiple-of-both-6-and-10-we-compute-the-least-common-multiple-lcm-of-6-and-10-33261492026-04-22T02:43:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sine-of--theta--equals--fracsqrt32--at--60circ--and--120circ--in-the-interval--0circ-360circ--these-angles-correspond-to-the-reference-angle--60circ--in-the-first-and-second-quadrants-thus-the-solutions-are--boxed60circ--and--boxed120circ--60150182026-04-22T03:02:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero--2x---13--0--solving--2x---3--0--gives--x--frac32--the-ordered-pair-is--boxedleft-frac32-3-right--963422026-04-22T03:02:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-cosine-function-oscillates-between---1--and--1--so--5cosleftfracpi6xright--ranges-from---5--to--5--adding-10-shifts-the-range-to--5-15--the-range-is--boxed5-15--2476612026-02-24T20:42:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--theta--arccosleftfrac1sqrt2right--so--cos-theta--frac1sqrt2--then--theta--45circ--and--tan-theta--1--the-answer-is--boxed1--14595162026-04-22T03:02:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-expression-under-the-square-root-must-satisfy--4---x2-geq-0--so--x-in--2-2--the-maximum-value-of--sqrt4---x2--is--2--at--x--0--and-the-minimum-is--0--at--x--pm-2--the-range-is--boxed0-2-think-43644782026-02-24T20:42:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-solve-for-mathbfv-we-use-the-cross-product-equation-mathbfv-times-mathbfw--mathbfp-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-calculated-as-beginpmatrix-v2-cdot-3---v3-cdot--1--v3-cdot-2---v1-cdot-3--v1-cdot--1---v2-cdot-2-endpmatrix--beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix-setting-this-equal-to-mathbfp--beginpmatrix-5--0---2-endpmatrix-we-get-the-system-11779142026-02-24T20:42:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix--beginpmatrix-3--0---2-endpmatrix-this-gives-the-system-44143282026-04-22T03:02:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-12-and-18-factorize-both-12--22-cdot-3-and-18--2-cdot-32-the-lcm-takes-the-highest-powers-of-all-primes-22-cdot-32--4-cdot-9--36-thus-the-lcm-is-boxed36-95351352026-04-22T03:08:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-minimum-depth-where-both-measurements-coincide-is-the-lcm-of-5-and-7-since-5-and-7-are-prime-lcm5-7--5-cdot-7--35-the-answer-is-boxed35-33704232026-04-22T03:08:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-arithmetic-mean-is-calculated-by-summing-the-values-and-dividing-by-the-count-frac10--14--163--frac403-approx-13overline3-the-exact-average-is-boxeddfrac403-25805582026-04-22T03:08:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-sum-the-expressions-4y---3--2y--5--y--7--7y--9-divide-by-3-and-set-equal-to-10-frac7y--93--10-multiply-both-sides-by-3-7y--9--30-subtract-9-7y--21-divide-by-7-y--boxed3-87885712026-04-22T03:08:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-cosine-of-45circ-is-given-by-cos-45circ--fracsqrt22-for-a-square-with-side-length-s--7-units-the-length-of-the-diagonal-d-can-be-calculated-using-the-pythagorean-theorem-d--ssqrt2-substituting-the-value-of-s-we-get-d--7sqrt2-therefore-the-length-of-the-diagonal-is-boxed7sqrt2-82004012026-04-22T03:10:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-a-of-a-regular-hexagon-with-side-length-s-is-given-by-the-formula-a--frac3sqrt32s2-setting-this-equal-to-54sqrt3-we-have-67402982026-04-22T03:10:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-a-of-the-triangle-is-given-by-93269302026-04-22T03:10:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-v-of-a-cone-is-given-by-29220642026-04-22T03:10:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-an-equilateral-triangle-with-side-length-s-the-radius-r-of-the-circumscribed-circle-is-given-by-54538292026-04-22T03:10:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--sqrtx--y--so--x--y2--substituting-into-the-equation-gives-61646682026-04-22T03:10:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-analyze-the-inequality--x--y--x---y-leq-8--this-expression-is-symmetric-and-represents-a-diamond-rhombus-centered-at-the-origin-2981362026-04-22T03:10:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--fx--ax3--bx2--cx--d--use-the-given-values-to-form-a-system-of-equations-52447132026-04-22T03:10:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--sqrtx--y--so--x--y2--substituting-gives-91098492026-04-22T03:11:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-one-marker-is-mandatory-choose-1-additional-marker-from-the-remaining-3-dbinom31-then-select-2-sequences-from-6-dbinom62-multiply-these-dbinom62-times-dbinom31--15-times-3--45-the-total-number-of-valid-combinations-is-boxed45question-a-tech-startup-founder-has-8-potential-investment-opportunities-and-wants-to-diversify-by-selecting-5-but-two-specific-opportunities-are-mutually-exclusive-how-many-valid-investment-portfolios-can-be-formed-81472902026-04-22T03:20:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-surjective-function-problem-count-the-number-of-ways-to-partition-7-distinct-species-into-4-non-empty-habitats-using-the-inclusion-exclusion-principle--sumk04--1k-binom4k-4---k7--calculating-47---binom4137--binom4227---binom4317-compute-each-term-16384---4-times-2187--6-times-128---4-times-1--16384---8748--768---4--8400-the-final-answer-is-boxed8400question-31920922026-02-24T21:00:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-tan60circ--fractextheighttextshadow-length-we-substitute-tan60circ--sqrt3-and-shadow-length-5textcm-solving-for-height-textheight--5-cdot-sqrt3-thus-boxed5sqrt3-67543952026-04-22T03:20:27+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-of-an-equilateral-triangle-is-fracsqrt34-s2-solving-for-original-side-s-fracsqrt34-s2--25sqrt3-rightarrow-s2--100-rightarrow-s--10textcm-new-side-14textcm-new-area-fracsqrt34-cdot-142--49sqrt3-increase-49sqrt3---25sqrt3--24sqrt3-thus-boxed24sqrt3-60896312026-04-22T03:20:27+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--h--represent-the-number-of-hours-worked-the-total-fee-for-the-first-consultant-is--300--75h--and-for-the-second-consultant-it-is--200--90h--setting-these-equal-2430282026-04-22T03:27:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--dt--kt2--5t--and-that--d3--48--substituting--t--3--74454282026-04-22T03:28:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-altitude-is-modeled-by-the-quadratic-function--y--3x2---12x--15--since-the-coefficient-of--x2--is-positive-the-parabola-opens-upward-and-the-minimum-value-occurs-at-the-vertex-the-x-coordinate-of-the-vertex-is-1962692026-04-22T03:28:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--e--represent-the-number-of-employees-the-total-cost-for-plan-a-is--100--10e--and-for-plan-b-it-is--150--7e--setting-these-equal-82992322026-04-22T03:28:05+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given--a--2b--substitute-into-the-expression-for--e--49837532026-04-22T03:28:05+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-function--ht---5t2--30t--10--is-a-quadratic-with-a-negative-leading-coefficient-so-the-maximum-occurs-at-the-vertex-the-time--t--at-which-the-maximum-altitude-occurs-is-99781832026-04-22T03:28:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-field-is-120-meters-wide-short-side-and-160-meters-long-long-side-to-ensure-full-coverage-the-drone-flies-parallel-passes-along-the-120-meter-width-with-each-pass-covering-20-meters-in-the-160-meter-direction-the-number-of-passes-required-is-frac12020--6-passes-each-pass-spans-160-meters-in-length-since-the-drone-turns-at-the-end-of-each-pass-and-flies-back-along-the-return-path-each-pass-contributes-160--160--320-meters-of-travelexcept-possibly-the-last-one-if-it-doesnt-need-to-return-but-since-every-pass-must-be-fully-flown-and-aligned-the-drone-must-complete-all-6-forward-and-6-reverse-segments-however-the-problem-states-it-aligns-passes-to-scan-fully-implying-the-drone-flies-each-pass-and-returns-so-6-forward-and-6-backward-segments-but-optimally-the-return-can-be-integrated-into-flight-planning-however-since-no-overlap-or-efficiency-gain-is-mentioned-assume-each-pass-is-a-continuous-straight-flight-and-the-return-is-part-of-the-route-but-standard-interpretation-for-full-coverage-with-back-and-forth-there-are-6-forward-passes-and-5-returns-noproblem-says-to-fully-scan-with-aligned-parallel-passes-suggesting-each-pass-is-flown-once-in-20m-width-and-the-drone-flies-each-160m-segment-and-the-turn-around-is-inherent-but-to-minimize-total-distance-assume-the-drone-flies-each-160m-segment-once-in-each-direction-per-pass-that-would-be-inefficient-but-in-precision-agriculture-standard-for-120m-width-6-passes-at-20m-width-the-drone-flies-6-successive-160m-lines-and-at-the-end-turns-and-flies-back-along-the-return-pathtypically-the-return-is-not-part-of-the-scan-but-the-drone-must-complete-the-loop-however-in-such-problems-its-standard-to-assume-each-parallel-pass-is-flown-once-in-each-direction-unlikely-better-interpretation-the-drone-flies-6-passes-of-160m-each-aligned-with-the-120m-width-and-the-return-from-the-far-end-is-not-counted-as-flight-since-its-typical-in-grid-scanning-but-problem-says-shortest-total-distance-so-we-assume-the-drone-must-make-6-forward-passes-and-must-return-to-start-for-safety-or-data-sync-so-6-forward-and-6-return-segments-each-160m-so-total-distance-6-times-160-times-2--1920-meters-but-is-the-return-160m-yes-if-flying-parallel-but-after-each-pass-it-returns-along-a-straight-line-parallel-so-160m-so-total-6-times-160-times-2--1920-but-waitcould-it-fly-return-at-angles-no-efficient-is-straight-back-but-another-optimization-after-finishing-a-pass-it-doesnt-need-to-turn-180--it-can-resume-along-the-adjacent-160m-segment-no-because-each-160m-segment-is-a-new-parallel-line-aligned-perpendicular-to-the-width-so-after-flying-north-on-the-first-pass-it-turns-west-180-to-fly-south-return-but-thats-still-160m-so-each-full-cycle-pass--return-is-320m-but-6-passes-require-6-returns-only-if-each-turn-around-is-a-complete-180-and-160m-straight-line-but-after-the-last-pass-it-may-not-need-to-returnit-finishes-but-problem-says-to-fully-scan-the-field-and-aligned-parallel-passes-so-likely-it-plans-all-6-passes-each-160m-and-must-complete-them-but-does-it-imply-a-return-the-problem-doesnt-specify-a-landing-or-reset-so-perhaps-the-drone-only-flies-the-6-passes-each-160m-and-the-return-flight-is-avoided-since-its-already-at-the-far-end-but-to-be-safe-assume-the-drone-must-complete-the-scanning-path-with-back-and-forth-turns-between-passes-so-6-upward-passes-160m-each-and-5-downward-returns-160m-each-totaling-6-times-160--5-times-160--11-times-160--1760-meters-but-standard-in-robotics-for-grid-coverage-total-distance-is-number-of-passes-times-width-times-2-forward-and-backward-but-only-if-returning-to-start-however-in-most-such-problems-unless-stated-otherwise-the-return-is-not-counted-beyond-the-scanning-legs-but-here-it-says-shortest-total-distance-so-efficiency-matters-but-no-turn-cost-given-so-assume-only-flight-distance-matters-and-the-drone-flies-each-160m-segment-once-per-pass-and-the-turn-between-is-instantso-total-flight-is-the-sum-of-the-6-passes-and-6-returns-only-if-full-loop-but-that-would-be-12-segments-of-160m-noeach-pass-is-160m-and-there-are-6-passes-and-between-each-a-return-that-would-be-6-passes-and-11-returns-no-clarify-the-drone-starts-flies-160m-for-pass-1-east-then-turns-west-180-flies-160m-return-back-then-turns-north-90-flies-160m-pass-2-etc-but-each-return-is-not-along-the-next-passeach-new-pass-is-a-new-160m-segment-in-a-perpendicular-direction-but-after-pass-1-east-to-fly-pass-2-north-it-must-turn-90-left-but-the-flight-path-is-now-160m-northso-its-a-corner-the-total-path-consists-of-6-segments-of-160m-each-in-consecutive-perpendicular-directions-forming-a-spiral-like-outer-loop-but-actually-orthogonal-the-path-is-160m-east-160m-north-160m-west-160m-south-etc-forming-a-rectangular-path-with-6-sides-no6-parallel-lines-alternating-directions-but-each-line-is-160m-and-there-are-6-such-lines-3-pairs-of-opposite-directions-the-return-between-lines-is-instantaneous-in-2dso-only-the-6-flight-segments-of-160m-matter-but-thats-not-realistic-in-reality-moving-from-the-end-of-a-160m-east-flight-to-a-160m-north-flight-requires-a-90-turn-but-the-distance-flown-is-still-the-160m-of-each-leg-so-total-flight-distance-is-6-times-160--960-meters-for-forward-plus-no-returnsince-after-each-pass-it-flies-the-next-pass-directly-but-to-position-for-the-next-pass-it-turns-but-that-turn-doesnt-add-distance-so-total-directed-flight-is-6-passes--160m--960m-but-is-that-sufficient-the-problem-says-to-fully-scan-so-each-120m-wide-strip-must-be-covered-and-with-6-passes-of-20m-width-its-done-and-aligned-with-shorter-side-so-minimal-path-is-6--160--960-meters-but-waitafter-the-first-pass-east-it-is-at-the-far-west-of-the-120m-strip-then-flies-north-for-160mthis-covers-the-north-end-of-the-strip-then-to-fly-south-to-restart-westward-it-turns-and-flies-160m-south-return-covering-the-south-end-then-east-etc-so-yes-each-160m-segment-aligns-with-a-new-120m-wide-parallel-and-the-160m-length-covers-the-entire-160m-span-of-that-direction-so-total-scanned-distance-is-6-times-160--960-meters-but-is-there-a-return-the-problem-doesnt-say-the-drone-must-return-to-startjust-to-fully-scan-so-960-meters-might-suffice-but-typically-in-such-drone-coverage-a-full-scan-requires-returning-to-begin-the-next-strip-but-here-no-indication-moreover-6-passes-of-160m-each-aligned-with-120m-width-fully-cover-the-area-so-total-flight-6-times-160--960-meters-but-earlier-thought-with-returns-was-incorrectno-separate-returnline-the-flight-is-continuous-with-turns-so-total-distance-is-960-meters-but-lets-confirm-dimensions-field-120m-w--160m-n-each-pass-160m-n-or-s-covering-a-120m-wide-band-6-passes-every-20m-covers-0120m-w-each-at-20m-intervals-020-2040--100120-each-pass-covers-one-120m-wide-strip-the-length-of-each-pass-is-160m-the-length-of-the-field-so-yes-6--160--960m-but-is-there-overlap-in-dense-grid-usually-offset-but-here-no-mention-of-offset-so-possibly-overlapping-but-for-minimum-distance-we-assume-no-redundancyoptimize-path-but-the-problem-doesnt-say-it-can-skip-turnsso-we-assume-the-optimal-path-is-6-straight-segments-of-160m-each-in-a-new-43191262026-02-24T21:08:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-dominant-to-recessive-traits-in-species-a-is--53--this-means-that-for-every-5-dominant-traits-there-are-3-recessive-traits-18116922026-04-22T03:33:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-algae-doubles-in-area-every-3-days-starting-from--87203122026-04-22T03:33:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-red-to-yellow-flowers-is--47--98224752026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-dna-segment-repeats-every-12-base-pairs-and-mutations-occur-every-48-base-pairs-6748702026-04-22T03:33:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-active-compound-to-dried-herb-is--33108382026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-annual-rainfall-is-60-inches-84182172026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio--31--means-3-out-of-every-4-plants-are-tall-and-1-out-of-every-4-is-short-83864592026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-distances-be--a---2d-a---d-a-a--d-a--2d--77152422026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-from-the-formula--an--3n--k--76474132026-04-22T03:33:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-fossils-found-at-the-five-sites-be-a---2d-a---d-a-a--d-and-a--2d-where-a-is-the-middle-term-and-d-is-the-common-difference-of-the-arithmetic-sequence-the-sum-of-these-terms-is-77548942026-04-22T03:44:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-original-number-of-bees-be-7x-and-butterflies-be-4x-after-adding-18-bees-the-new-number-of-bees-is-7x--18-the-new-ratio-is-given-as-53-so-13925672026-04-22T03:44:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-note-the-only-even-prime-is--2--all-other-primes-are-odd-the-sum-of-three-numbers-is-even-if-38098872026-04-22T03:55:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-smallest-four-digit-number-divisible-by-both-7-and-13-we-first-compute-their-least-common-multiple-since-7-and-13-are-both-prime-their-lcm-is-simply-their-product-50641202026-04-22T04:08:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-simplify-reduce-each-base-modulo-8-77395742026-04-22T04:08:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--their-product-is-57579132026-04-22T04:08:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-point--d--such-that--abcd--is-a-parallelogram-we-use-the-property-that-in-a-parallelogram-the-diagonals-bisect-each-other-therefore-the-midpoint-of-diagonal--ac--must-be-the-same-as-the-midpoint-of-diagonal--bd--24658852026-04-22T04:15:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-total-volume-of-nutrient-solution-required-multiply-the-amount-needed-per-culture-by-the-number-of-cultures-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utcomes-is-69612842026-04-22T07:03:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-shortest-altitude-in-a-triangle-with-side-lengths--a--13---b--14--and--c--15--we-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--is-19431832026-04-24T12:56:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-rectangle-with-dimensions-5-cm-by-12-cm-is-inscribed-in-a-circle-so-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-66484732026-04-24T12:56:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-a-right-triangle-with-legs--a---b--hypotenuse--z--25--and-inradius--c--5--we-use-the-formula-for-the-inradius-of-a-right-triangle-16788812026-04-24T12:56:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-an-equilateral-triangle-with-side--s--is-80478252026-04-24T12:56:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-base-and-each-leg-have-length--x--then-the-perimeter-is--x--x--x--3x--36--so--x--12--67006172026-04-24T12:56:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-determine-the-prime-factorization-of-4800-begin-by-factoring-step-by-step-53512892026-03-12T12:20:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--v--6k--where--k--is-a-positive-integer-then-2602652026-03-12T12:20:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-simplify-the-expression-82319812026-03-12T13:09:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-looking-for-the-number-of-integers-from-1-to-100-that-are-congruent-to-3-modulo-7-these-numbers-form-an-arithmetic-sequence-37352682026-04-24T12:58:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-be-of-the-form-1000a--10b--7-ie-ends-in-7-we-want-the-smallest-such-number-divisible-by-13-45584362026-04-22T07:20:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-side-length-of-the-square-be--s--the-diagonal-of-a-square-divides-it-into-two-congruent-right-triangles-and-the-relationship-between-the-side-length-and-the-diagonal-is-given-by-the-pythagorean-theorem-64409892026-04-22T07:20:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-let-the-sides-be--a--13---b--14---c--15--the-semi-perimeter-is-68271432026-04-22T07:20:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-radius-of-the-cylinder-be--r--and-its-height-be--h--2r--the-cylinder-is-inscribed-in-a-sphere-of-radius-10-so-the-diagonal-from-the-center-of-the-sphere-to-the-top-rim-of-the-cylinder-forms-a-right-triangle-with-legs--r--and--r--and-hypotenuse-equal-to-the-radius-of-the-sphere-10-cm-71982412026-04-22T07:20:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-number-of-6-digit-binary-numbers-with-exactly-one-11-block-we-analyze-each-possible-position-of-the-11-and-ensure-no-additional-consecutive-1s-the-11-can-occupy-positions-12-23-34-45-or-56-for-each-case-the-remaining-digits-must-avoid-forming-additional-11-blocks-82871182026-04-22T07:24:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-the-number-of-onto-functions-from-5-species-to-3-zones-calculated-using-inclusion-exclusion-3127682026-04-22T07:23:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-shoelace-formula-for-the-area-of-a-triangle-with-vertices--x1-y1---x2-y2---x3-y3--93716332026-04-22T07:24:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-two-complex-numbers--z1--x1--y1i--and--z2--x2--y2i--are-orthogonal-if-their-dot-product--x1x2--y1y2--0--70842762026-04-22T07:24:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-variance-of-a-dataset-is-defined-as-the-average-of-the-squared-deviations-from-the-mean-given-that-there-are-12-samples-and-the-sum-of-the-squares-of-the-deviations-is-144-we-compute-the-variance-as-follows-95044992026-04-22T07:24:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-function--pd--3d--7--and-we-are-to-compute-the-total-number-of-microplastic-particles-from-depth--d--1--to--d--10--this-requires-summing--pd--over-integer-values-from-1-to-10-2397652026-04-22T07:24:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-first-term-of-the-arithmetic-sequence-be--a--and-the-common-difference-be--d--the-sequence-has-8-terms-so-the-sum--s8--is-66653832026-04-22T07:24:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-asked-to-count-the-number-of-distinct-sequences-in-which-the-drone-can-visit-5-fields-2-wheat-w-2-corn-c-and-1-soybean-s-where-fields-of-the-same-crop-are-indistinguishable-this-is-a-permutation-of-a-multiset-37506362026-04-24T13:00:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-9-nucleotides-4-as-3-cs-and-2-gs-where-nucleotides-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-22168052026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-6-days-to-cultivation-tasks-3-wheat-w-2-corn-c-and-1-soybean-s-where-the-order-matters-but-tractors-are-indistinct-the-number-of-distinct-sequences-is-the-multinomial-coefficient-33947382026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-5-proteins-2-as-2-gs-and-1-f-with-indistinguishable-items-within-each-type-the-number-of-distinct-sequences-is-24249672026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-asked-to-count-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-icons-is-4--3--2--9-with-repetitions-of-each-type-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-935952026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-8-individuals-with-repetitions-3-chimpanzees-2-gorillas-and-3-orangutans-the-number-of-distinct-arrangements-is-84044592026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-10-items-with-repetitions-5-of-one-type-3-of-another-and-2-of-a-third-the-number-of-distinct-sequences-is-78034262026-04-24T13:00:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-9-items-with-repetitions-4-males-3-females-2-juveniles-the-number-of-distinct-sequences-is-13114152026-04-22T07:24:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-average-we-add-the-three-expressions-and-divide-by-3-50870312026-04-24T13:01:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-in-a-regular-icosahedron-all-edges-are-of-equal-length-first-compute-the-distance-between-the-given-points-32997652026-04-24T13:01:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-solve-this-we-use-combinations-to-count-favorable-outcomes-over-total-outcomes-93265142026-04-24T13:01:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-each-altitude-has-one-sensor-selecting-3-sensors-with-no-two-from-the-same-altitude-simply-means-choosing-any-3-distinct-sensors-which-is-guaranteed-as-all-sensors-are-at-different-altitudes-14510122026-04-24T13:01:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-robot-arms-are-identical-the-problem-reduces-to-choosing-4-distinct-arms-out-of-6-to-place-the-sensors-however-because-the-arms-are-indistinct-the-order-of-placement-does-not-matter-thus-the-number-of-ways-is-the-number-of-combinations-of-6-arms-taken-4-at-a-time-41657522026-04-24T13:01:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-equivalent-to-partitioning-7-distinct-calculators-into-4-non-empty-identical-subsets-the-number-of-ways-is-given-by-the-stirling-numbers-of-the-second-kind--s7-4--using-the-formula-or-recurrence-relation-94639332026-04-24T13:01:49+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-four-terms-of-the-arithmetic-sequence-be-a---3d-a---d-a--d-and-a--3d-where-a-is-the-middle-term-and-d-is-the-common-difference-14630772026-04-22T07:37:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diagonal-of-the-rectangle-is-the-hypotenuse-of-a-right-triangle-with-legs-5-cm-and-12-cm-using-the-pythagorean-theorem-14113852026-04-22T07:43:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-data-entries-form-an-arithmetic-sequence-an--1--3n---1-for-n--1-to-100-the-sum-is-s--75143772026-04-22T07:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-recursive-sequence-95864182026-04-22T07:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-arrangements-of-9-symbols-with-3-identical-is--77822612026-04-24T13:07:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-first-five-prime-numbers-are-2-3-5-7-and-11-since-these-numbers-are-all-prime-their-least-common-multiple-is-simply-their-product-52948922026-04-24T13:07:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-observe-that-all-the-numbers-are-odd-and-consecutive-odd-numbers-we-can-reduce-each-number-modulo-8-79620012026-04-24T13:07:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-population-doubles-every-3-hours-we-can-model-the-growth-using-exponential-growth-88648042026-04-22T07:59:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-length-be--l--and-the-width-be--w--the-perimeter-is-given-by-46656872026-04-22T07:59:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-looking-for-the-least-common-multiple-lcm-of-12-and-18-which-represents-the-first-day-both-strains-share-the-mutation-again-82388672026-04-22T07:59:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-the-first-component-to-the-second-is-53-let--x--be-the-amount-of-the-second-component-then-4968712026-04-22T07:59:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-length-be--l--and-the-width-be--w--the-area-is-given-by-50290182026-04-22T07:59:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-an-arithmetic-progression-with-first-term--a--3--and-common-difference--d--4--the-general-term-of-an-arithmetic-sequence-is-given-by-53772282026-04-22T07:59:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-asked-to-compute-the-sum-of-the-squares-of-the-first--7--odd-numbers-modulo--10--the-first--7--odd-numbers-are--1-3-5-7-9-11-13--their-squares-are-37515362026-04-22T08:04:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-convert-from-base-eight-to-base-ten-expand-the-number-as-a-sum-of-powers-of-8-24015642026-04-22T08:08:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-formula-for-the-sum-of-squares-86986742026-04-22T08:08:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-from-trigonometric-values-81215052026-04-22T08:17:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-greatest-common-thickness-that-divides-all-measurements-we-compute-the-greatest-common-divisor-gcd-of-all-four-thicknesses-360-480-270-and-540-68263912026-04-22T08:20:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-arithmetic-mean-is-calculated-by-summing-the-transactions-and-dividing-by-the-number-of-departments-74905372026-04-22T08:20:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-sequences-align-we-compute-the-least-common-multiple-lcm-of-18-and-24-31570752026-04-22T08:20:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-length-is-calculated-by-11205432026-04-22T08:20:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-problem-involves-partitioning-15-distinct-grants-into-5-groups-of-3-each-where-the-order-of-grants-within-a-university-does-not-matter-the-number-of-ways-is-given-by-the-multinomial-coefficient-18940372026-03-12T22:35:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-combinatorial-problem-with-adjacency-constraints-we-model-the-grid-as-a-4x4-lattice-where-each-cell-is-p-or-i-and-no-two-horizontally-or-vertically-adjacent-cells-are-both-p-this-is-equivalent-to-counting-binary-matrices-with-no-two-adjacent-1s-p--1-i--0-76786212026-03-12T22:34:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-equivalent-to-partitioning-6-distinct-compounds-into-3-non-empty-unordered-subsets-phases-the-number-of-ways-is-the-stirling-number-of-the-second-kind--s6-3--multiplied-by-the-number-of-ways-to-assign-these-subsets-to-3-distinguishable-phases-since-phases-are-distinct-eg-phase-1-2-3-83816012026-03-12T22:36:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-10-distinguishable-countries-into-3-categories-with-specified-counts-4-in-highly-automated-3-in-moderately-affected-and-the-remaining-3-in-minimally-influenced-13721692026-04-24T13:19:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-9-sensors-where-4-are-ai-based-indistinguishable-among-themselves-3-are-traditional-indistinguishable-among-themselves-and-2-are-hybrid-indistinguishable-among-themselves-73407952026-04-24T13:19:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-7-distinguishable-tribes-into-3-categories-with-fixed-counts-3-agrarian-2-nomadic-and-2-maritime-70762522026-04-24T13:19:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-12-distinguishable-job-roles-into-3-categories-with-fixed-counts-5-ai-driven-4-enhanced-by-ai-and-3-unaffected-4733362026-04-24T13:19:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-8-events-where-3-are-strong-3-are-moderate-and-2-are-weak-since-events-of-the-same-type-are-indistinguishable-the-number-of-sequences-is-76674572026-04-24T13:19:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-9-distinguishable-artifacts-into-3-time-periods-with-fixed-counts-4-ancient-3-classical-and-2-modern-35213952026-04-24T13:19:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--m--be-the-midpoint-of--ab--and--n--be-the-midpoint-of--ac--then-44253892026-04-24T13:19:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-partial-fraction-decomposition-72062872026-04-22T08:53:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-difference--fx1---fx--is-linear--fx--must-be-a-quadratic-47060672026-04-22T08:53:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-radius--r--of-the-circumscribed-circle-of-a-triangle-with-sides--a--5---b--6--and--c--7--we-use-the-formula-69645952026-04-22T08:54:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-a-triangle-can-be-calculated-using-the-bas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4-22T09:21:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-by-expanding-both-sides-491672026-04-22T09:21:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-average-add-the-two-values-and-divide-by-2-57812522026-04-22T09:22:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-by-expanding-the-left-side-20492322026-04-22T09:32:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-length-is-found-by-adding-the-lengths-and-dividing-by-2-53951592026-04-22T09:32:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-in-a-regular-tetrahedron-all-edges-are-equal-compute-the-distance-between-given-points-98121712026-04-22T09:32:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-looking-for-the-smallest-three-digit-number-n-such-that-11178142026-04-22T09:33:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-point-on-the-line--y--72304482026-04-22T09:40:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-three-given-vertices-be--a100---b010---c001--and-let--dxyz--be-the-fourth-vertex-of-the-regular-tetrahedron-with-integer-coordinates-in-a-regular-tetrahedron-all-edges-are-equal-13480272026-04-24T13:38:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--c--13--the-radius--r--of-the-incircle-of-a-right-triangle-is-given-by-46900692026-04-22T09:42:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-5-character-strings-using-the-letters-a-b-c-d-4-letters-such-that-each-letter-appears-at-least-once-this-is-a-classic-inclusion-exclusion-problem-34399382026-04-22T09:43:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diagonal-of-a-rectangle-forms-a-right-triangle-with-its-length-and-width-using-the-pythagorean-theorem-the-diagonal-d-is-calculated-as-48238932026-04-22T09:43:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-in-a-regular-hexagon-inscribed-in-a-circle-the-radius-equals-the-side-length-of-the-hexagon-this-is-because-the-hexagon-can-be-divided-into-six-equilateral-triangles-thus-the-radius-is-equal-to-the-side-length-34742692026-04-22T09:43:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-center-complete-the-square-for-both--x--and--y--68405802026-04-22T09:43:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-two-vectors-to-be-perpendicular-their-dot-product-must-be-zero-53566162026-04-24T13:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is-given-by-99433242026-04-24T13:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-formula-for-the-inradius-21591732026-04-24T13:41:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-a-team-of-4-experts-from-7-mathematicians-5-physicists-and-6-computer-scientists-with-the-restriction-that-at-least-one-expert-must-come-from-each-discipline-31807252026-04-24T13:42:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-5-distinct-people-around-a-circular-table-where-rotations-are-considered-identical-in-circular-permutations-the-number-of-arrangements-of--n--distinct-people-is--n-1--64802562026-04-24T13:42:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-forming-a-6-digit-code-with-the-following-constraints-99570442026-04-24T13:42:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-a-6-member-committee-from-10-scientists-including-3-ai-experts-and-8-engineers-with-40901692026-04-22T09:53:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-closest-point-on-a-line-to-a-given-point-lies-along-the-perpendicular-from-the-point-to-the-line-the-slope-of-the-given-line-is----4077842026-04-24T13:46:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-radius-of-the-largest-circle-that-fits-inside-the-triangle-the-incircle-we-use-the-formula-96029072026-04-24T13:46:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-surface-area-of-a-cone-includes-the-lateral-curved-surface-area-and-the-base-area-the-formula-for-total-surface-area-is-30407892026-04-22T10:07:07+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-angle-of-elevation--heta--can-be-found-using-the-tangent-function-38349842026-04-24T13:46:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume--v--of-a-regular-tetrahedron-with-edge-length--a--is-14058992026-04-24T13:46:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-width-be--w--meters-and-the-length-be--3w--meters-the-perimeter-is-given-by-21493822026-04-24T13:46:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-2000-2004-2008--2024-this-is-an-arithmetic-sequence-with-first-term--a--2000--common-difference--d--4--and-last-term--l--2024--the-number-of-terms--n--is-given-by-92529832026-04-24T13:49:06+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-remainder-when-the-sum-of-the-patent-numbers-20250414--20250608--20250702--20250796-is-divided-by-100-we-only-need-the-last-two-digits-of-each-number-96311742026-04-22T10:15:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sum-of-the-squares-of-the-first-n-positive-integers-is-given-by-the-formula-36663772026-04-22T10:15:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-both-sides-by-10-to-eliminate-denominators-61452752026-04-22T10:21:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the--y--intercept-occurs-when--x--0--substitute--x--0--into-the-equation-60744552026-04-22T10:21:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-terms-be--a---2d-a---d-a-a--d-a--2d--the-fifth-term-is--a--2d--and-the-first-is--a---2d--their-sum-is-74767422026-04-22T10:24:15+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-the-expressions-equal-98751582026-04-22T10:24:15+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-distributive-property-foil-method-85987752026-04-22T10:28:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-width-of-the-rectangle-be--w--meters-then-the-length-is--3w--meters-the-perimeter-is-given-by-18291502026-04-24T13:54:27+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-weights-of-the-two-foxes-be--x--and--y--we-are-given-64737142026-04-22T10:37:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-arithmetic-with-first-term--a1--3--and-common-difference--d--5--the--n--th-term-is-given-by--an--a1--n-1d--for--n--7--32066792026-04-22T10:45:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--t--3--into-the-equation-81429512026-04-22T10:45:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-geometric-with--a1--1---r--2--the-sum-of--n--5--terms-is-48244512026-04-22T10:45:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-how-many-4-digit-numbers-are-divisible-by-11-79349822026-04-22T10:49:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-robots-net-position-subtract-the-backward-distance-from-the-forward-distance-81816682026-04-22T10:49:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-average-length-add-the-two-lengths-and-divide-by-2-93792602026-04-22T10:49:05+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-smallest-5-digit-number-is--10000--the-largest-is--99999--45183832026-04-22T10:49:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-the-triangle-is-92771902026-04-22T10:54:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-elements-is-6--4--3--13-since-items-within-each-category-are-indistinguishable-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-85253922026-04-22T10:58:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-11-positions-where-5-are-clay-c-2-are-silt-s-and-4-are-loam-l-with-indistinguishability-within-each-type-the-number-of-distinct-sequences-is-the-multinomial-coefficient-71313272026-04-22T10:58:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-there-are-5--3--2--10-days-total-the-number-of-distinct-sequences-is-given-by-75260982026-04-22T10:58:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-devices-is-4--3--3--10-with-indistinguishability-within-each-category-the-number-of-distinct-sequences-is-46075912026-04-22T10:58:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-15-items-9-identical-observations-o-4-identical-thought-experiments-t-and-2-identical-historical-cases-h-the-number-of-distinct-sequences-is-7697142026-04-22T10:58:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-count-the-number-of-valid-lattice-paths-from-0-0-to-5-4-that-do-not-cross-the-line--y--x--1--we-use-the-reflection-principle-first-compute-the-total-number-of-unrestricted-paths-from-0-0-to-5-4-25340022026-04-22T11:00:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-intersection-set-the-two-expressions-for--y--equal-19946302026-04-22T11:02:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--t--represent-the-number-of-hours-after-which-the-volumes-are-equal-64234982026-04-24T14:03:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-two-numbers-be--a--and--b--we-are-given-82528322026-04-22T11:15:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-two-variables-be--a--and--b--we-are-given-78133002026-04-22T11:15:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-the-growth-rate-i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urces/solution-the-sequence-is-arithmetic-with-first-term--a1--5--common-difference--d--4--and-last-term--an--41--58127382026-04-22T12:17:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-up-the-equation-for-the-average-of-the-two-expressions-31942292026-04-24T14:23:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-number-of-migration-events-divide-the-total-days-by-the-cycle-length-5798142026-04-24T14:23:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-least-common-multiple-lcm-of-8-and-12-determines-when-both-events-align-prime-factorizations-44523012026-04-24T14:23:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-lcm-of-6-and-9-is-calculated-as-follows-67998392026-04-24T14:23:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-desired-integer-be--x--79662442026-04-22T12:23:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-blue-markers-be-x-the-ratio-54-implies-25708472026-04-24T14:25:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-mean-is-calculated-by-summing-the-values-and-dividing-by-2-96070882026-04-24T14:25:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-reactions-are-independent-the-probability-that-all-three-succeed-is-77837552026-04-24T14:25:24+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-n--4-so-the-deck-has-3n--12-cards-labeled-from-1-to-12-we-want-the-probability-that-the-sum-of-3-randomly-drawn-distinct-cards-is-divisible-by-3-95162442026-04-24T14:25:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--1--into-both-functions-62736992026-04-22T12:28:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-the-equations-equal-at--x--3--46780632026-04-22T12:28:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-multinomial-coefficient-the-number-of-ways-is--86580272026-04-22T12:29:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume--v--of-a-prism-is-given-by-the-base-area--a--multiplied-by-the-height--h--thus--v-30818302026-04-24T14:29:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-is-calculated-as--87250402026-04-24T14:30:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-find-each-number-modulo-13-82476412026-04-24T14:30:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-first-team-has-4-prototypes-each-with-a-distinct-rank-from-1-to-4-the-third-team-has-3-prototypes-ranked-1-to-3-since-one-prototype-is-selected-uniformly-at-random-from-each-team-the-selection-within-each-team-is-independent-40882832026-04-22T12:45:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-a-total-of--5--2--1--8--atoms-where-5-are-identical-hydrogen-h-2-are-identical-oxygen-o-and-1-is-nitrogen-n-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-45849802026-04-22T12:45:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-greatest-common-factor-gcf-of-78-and-104-we-start-by-finding-their-prime-factorizations-58480602026-04-24T14:31:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-of-three-expressions-is-given-by-15298822026-04-24T14:31:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--t--15--into-the-equation-19300292026-04-22T12:52:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-lengths-be--3x---5x--and--7x--the-sum-is--3x--5x--7x--15x--45--solving-for--x--16167072026-04-22T12:52:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-rectangular-trench-is-inscribed-in-the-circular-foundation-the-diagonal-of-the-rectangle-is-equal-to-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-88390692026-04-22T12:52:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-rectangle-is-inscribed-in-the-circle-so-its-diagonal-is-the-diameter-the-diagonal-d-is-72753352026-04-22T12:52:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-simplify-the-expression--fx--46813082026-04-22T13:03:42+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-arithmetic-mean-sum-all-the-measurements-and-divide-by-the-number-of-days-83056242026-03-14T06:02:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-total-mixture--15--35--50-liters-fraction-of-rainwater---6433852026-03-14T06:03:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-there-are-only-two-types-high-income-and-low-income-it-is-impossible-to-select-3-zones-of-different-types-without-including-at-least-two-of-the-same-type-therefore-the-event-of-selecting-3-zones-all-of-different-types-is-impossible-9276322026-03-14T06:03:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-intersection-point-solve-the-system-of-equations-99197282026-03-14T06:03:45+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-nous-utilisons-la-formule-daddition-pour-la-tangente--55925022026-04-22T13:08:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-use-the-distributive-property-to-expand-22066862026-04-22T13:11:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-area-of-a-triangle-with-sides-a--7-b--10-and-c--13-we-use-herons-formula-99111982026-03-14T07:04:43+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero-compute-3-2--x5--0-this-simplifies-to--6--5x--0-solving-gives-x--69405922026-04-22T13:25:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-chips-is-5--7--3--15-the-probability-the-first-chip-is-amber-is--54788222026-04-22T13:26:01+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-shortest-altitude-corresponds-to-the-longest-side-8-cm-compute-area-using-herons-formula--s--59303472026-04-22T13:26:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-factor-1080-we-have-15039092026-04-24T14:43:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-looking-for-the-largest-number-n--200-such-that-91166062026-04-24T14:43:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-area-using-herons-formula-the-semiperimeter-is-1291532026-04-24T14:45:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given-hypotenuse--25--one-leg--24--the-other-leg-is-86936002026-04-24T14:45:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-problem-involves-distributing-4-distinct-objects-into-2-identical-boxes-this-corresponds-to-the-sum-of-stirling-numbers-of-the-second-kind-for-2-boxes-considering-non-empty-subsets-the-formula-is--s41--s42--here--s41--1--all-groups-in-one-zone-and--s42--7--split-into-two-non-empty-groups-thus-the-total-number-of-ways-is--1--7--8--24771522026-04-22T13:37:55+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-for-circular-arrangements-of-n-distinguishable-people-the-number-of-distinct-arrangements-is-n-1-here-there-are-8--5--13-individuals-thus-the-number-of-distinct-seating-arrangements-is-13-1--12-64797642026-04-22T13:38:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinguishable-objects-into-exactly-4-non-empty-indistinguishable-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-s6-4-from-known-values-s6-4--65-since-the-cabinets-are-indistinguishable-we-do-not-multiply-by-permutations-58253692026-04-22T13:38:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-an-arithmetic-sequence-where-the-first-term--a--3--the-common-difference--d--4--and-the-last-term--l--99--to-find-the-number-of-terms--n--we-use-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-64352992026-04-22T13:38:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-to-find-how-many-of-these-are-divisible-by-11-we-find-the-smallest-and-largest-multiples-of-11-in-this-range-48871702026-04-22T13:38:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-count-the-positive-divisors-of-60-the-prime-factorization-of-60-is-92399052026-04-22T13:38:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-center-rewrite-the-equation-by-completing-the-square-group-x-and-y-terms-21400812026-04-22T13:38:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-a-system-of-congruences-6789932026-03-14T10:18:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-3-toucans-t-2-tanagers-a-1-macaw-m-and-1-hummingbird-h-and-we-want-to-count-the-number-of-distinct-sequences-of-length-7-using-these-indistinguishable-items-the-total-number-of-distinct-permutations-of-a-multiset-is-given-by-69252822026-04-22T13:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-assigning-4-identical-robots-r-3-identical-detectors-d-and-2-identical-sensors-s-to-9-distinct-stations-with-one-device-per-station-this-is-a-multinomial-coefficient-problem-71215952026-04-22T13:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-choose-2-different-flavors-from-8-the-number-of-combinations-is-67188902026-04-22T13:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-2-jaguars-j-2-capybaras-c-and-2-sloths-s-all-indistinguishable-within-species-the-number-of-distinct-permutations-is-70147392026-04-22T13:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-distributing-5-distinguishable-objects-wires-into-3-indistinguishable-non-empty-boxes-this-is-equivalent-to-finding-the-stirling-numbers-of-the-second-kind-s53-which-counts-the-number-of-ways-to-partition-5-labeled-items-into-3-non-empty-unlabeled-subsets-from-known-values-21220292026-04-22T13:43:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-an-equilateral-triangle-with-side-length-6-km-and-we-are-to-find-the-perpendicular-distance-from-a-vertex-to-the-opposite-sidethis-is-the-altitude-of-the-equilateral-triangle-36200852026-04-22T13:43:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-triangle-has-side-lengths-a--13-b--14-c--15-the-longest-side-is-15-cm-so-the-shortest-altitude-corresponds-to-this-side-80986682026-04-22T13:43:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-of-the-sphere-is-73645532026-03-14T10:53:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-verify-the-triangle-is-right-angled-46093132026-03-14T10:53:44+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-convert--3427--to-base-ten-we-expand-it-using-powers-of-7-43204992026-04-22T13:53:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-7-and-11-are-coprime-their-least-common-multiple-is-57413012026-04-24T14:50:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-length-is-calculated-by-summing-the-lengths-and-dividing-by-2-95841472026-04-22T14:00:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-find-the-cost-of-one-sensor-96899762026-04-22T14:00:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-numerical-values-61238992026-04-22T14:00:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-total-cost-of-three-smartwatches-multiply-the-cost-of-one-by-3-21489222026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-average-add-the-three-expressions-and-divide-by-3-70438452026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-red-marbles-be-4x-and-blue-marbles-be-7x-since-4x--28-solve-for-x-99973062026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-amount-each-friend-pays-divide-the-total-cost-by-4-4808532026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-yellow-candies-be-5x-and-orange-candies-be-8x-since-8x--40-solve-for-x-88055112026-04-22T14:08:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-a-multiset-of-12-soil-cores-3-from-plot-a-2-from-plot-b-and-1-each-from-plots-c-d-e-and-f-the-total-number-of-distinct-sequences-arrangements-in-which-these-indistinct-items-can-be-arranged-is-a-multinomial-coefficient-61950862026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-distinct-sequences-of-12-blocks-where-there-are-5-red-r-4-blue-b-and-3-green-g-and-blocks-of-the-same-color-are-indistinguishable-this-is-a-permutation-of-a-multiset-65434612026-04-22T14:08:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-volume-of-a-sphere-with-radius--r--2y--74001182026-04-24T14:55:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-shortest-distance-from-a-point-x0-y0-to-a-line-defined-by-ax--by--c--0-is-given-by-the-formula-20395472026-04-22T14:14:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-two-area-functions-75421212026-04-22T14:14:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-first-term-be-a-and-the-common-difference-be-d-91299262026-04-22T14:14:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-number-of-ways-to-choose-3-compounds-from-8-and-2-delivery-mechanisms-from-5-we-use-combinations-85144872026-04-22T14:15:24+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-oak-pollen-grains-be--x--the-ratio--75--implies-51087812026-04-22T14:32:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-white-flowered-plants-be--y--the-ratio--53--implies-98803702026-04-22T14:32:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-dominant-monkeys-be--z--the-ratio--411--implies-3445542026-04-22T14:32:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-an-arithmetic-progression-28331042026-04-22T14:32:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-seating-is-circular-and-the-2-newtonian-experts-must-sit-together-we-treat-them-as-a-single-block-this-reduces-the-problem-to-arranging-6-units-the-block-and-5-historians-in-a-circle-the-number-of-circular-permutations-of-n-objects-is-n-1-so-we-fix-the-block-and-arrange-the-5-historians-around-it-giving-6---1--5-arrangements-within-the-block-the-2-newtonian-experts-can-be-arranged-in-2-ways-thus-the-total-number-of-arrangements-is-20358472026-04-22T14:32:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--c--be-coding-rate-and--t--testing-rate-the-system-is-93693842026-04-24T15:01:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-y-intercept-set--h--0--37460492026-04-24T15:01:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-expression-simplifies-to-75875132026-04-22T14:42:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given-the-average-detection-over-three-locations-is-20-ppm-we-set-up-the-equation-for-the-average-10822692026-04-22T14:46:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-processing-rate-is-given-by-56829772026-04-22T14:46:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-average-pressure-change-is-given-by-57695172026-04-22T14:46:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-kale-to-spinach-is-75-meaning-for-every-7-kale-plants-there-are-5-spinach-plants-since-there-are-35-kale-plants-we-determine-how-many-groups-of-7-are-in-35-30462432026-04-24T15:05:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-of-basil-to-mint-channels-is-43-so-for-every-4-basil-channels-there-are-3-mint-channels-since-there-are-28-basil-channels-we-calculate-the-number-of-groups-of-4-in-28-15894772026-04-24T15:05:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-1660-and-1666-factorize-both-numbers-35388182026-04-24T15:05:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-ratio-74-means-7-parts-wheat-correspond-to-4-parts-barley-54682412026-03-14T19:29:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-pour-trouver-la-valeur-minimale-de--53704362026-03-14T19:39:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area-of-the-path-is-the-difference-between-the-area-of-the-larger-circle-garden--path-and-the-garden-itself-the-radius-of-the-larger-circle-is--r--w--thus-the-area-of-the-path-is-46976252026-04-22T14:52:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-the-expression-equal-to-5-and-solve-for--x--38312562026-04-22T14:52:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--1--into-the-polynomial-74643242026-04-22T14:52:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--3--into-the-quadratic-equation-96509032026-04-22T14:52:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--6--into-the-function-91933102026-04-22T14:52:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--0--into-the-function-85841662026-04-22T14:52:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--4--into-the-function-31482272026-04-22T14:52:57+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-number-of-toucans-be--t--and-the-number-of-parrots-be--p--from-the-given-ratio-we-have-76316902026-04-22T15:03:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-formula-for-the-area-of-a-triangle-with-coordinates-58238882026-04-22T15:06:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-3-applications-at-random-without-replacement-from-120-total-and-we-want-the-probability-that-each-application-is-from-a-distinct-fieldone-from-cancer-one-from-neuroscience-and-one-from-infectious-diseases-46646722026-04-24T15:10:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-greatest-common-divisor-gcd-of-144-and-60-we-first-perform-the-prime-factorization-of-each-number-7570732026-04-22T15:07:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-find-the-cost-of-one-sensor-by-dividing-the-total-cost-by-the-number-of-sensors-62659922026-04-22T15:07:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-smallest-5-digit-number-is-10000-and-the-largest-is-99999-21486712026-04-22T15:07:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-area-of-the-square-24687232026-04-22T15:07:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is-arithmetic-with-first-term--a1--2--common-difference--d--3--and-last-term--an--50--using-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-82013932026-03-14T21:08:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-energy-after--n--months-is-the-sum-of-the-arithmetic-series-90333182026-03-14T21:08:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--x--2--and--y---1--into-the-expression-for--e--84681752026-03-14T21:08:08+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-start-by-recalling-the-identity-235712026-03-14T21:08:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x--be-the-integer-we-have-3873992026-04-22T15:16:25+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-total-distance-we-add-the-two-mixed-numbers-67100272026-04-22T15:17:09+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-total-flow-rate-add-the-two-mixed-numbers-28242022026-04-22T15:17:08+00:00daily0.8https://iqdig1.iqn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ind-the-greatest-common-factor-gcf-of-36-54-and-81-prime-factorizations-27620472026-04-22T16:16:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-law-of-cosines-to-find-the-largest-angle-opposite-the-longest-side-13-cm-73834392026-04-22T16:22:04+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diagonal-support-beam-stretches-from-one-point-on-the-edge-of-the-circular-base-through-the-center-to-the-opposite-point-so-its-length-is-the-diameter-79478782026-04-22T16:23:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-radius-r-of-the-circumscribed-circle-of-a-triangle-with-sides-a--50-b--60-and-c--70-we-use-the-formula-90504222026-04-22T16:23:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-radius-of-the-base-is-half-the-diameter-70453552026-04-22T16:23:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-triangle-with-sides-7-24-and-25-is-a-right-triangle-because-58694262026-04-22T16:23:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-average-sum-the-three-expressions-and-divide-by-3-4956312026-04-22T16:24:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-smallest-3-digit-number-divisible-by-7-is-105--7-imes-15-and-the-largest-is-994--7-imes-142-15788842026-04-22T16:24:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-sum-the-three-expressions-8b---1--5b--4--2b---7--15b---4-79550052026-04-22T16:24:58+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-concentrations-at-each-hour-are-93382162026-03-15T03:07:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-of-dates-in-bce-is-an-arithmetic-sequence-starting-at-1050-decreasing-by-5-ending-at-1000-75658932026-03-15T03:07:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-6897352026-03-15T03:07:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-98240462026-03-15T03:07:27+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-sequence-is--a-ad-a2d-a3d-a4d--the-middle-term-is--a2d--5--so-11413432026-04-22T16:38:22+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-apply-the-am-gm-inequality-88942282026-04-22T16:42:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-difference-gx--1---gx-is-linear-gx-must-be-a-quadratic-polynomial-let-26138782026-04-22T16:42:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-function-is-a-quadratic-with-a-minimum-at--x--50227542026-04-22T16:44:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-3-species-out-of-7-and-4-of-them-are-known-to-use-quantum-entanglement-based-communication-we-want-all-3-selected-species-to-belong-to-this-group-6352702026-04-22T16:58:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-rectangular-sensor-section-is-inscribed-within-the-circular-cross-section-of-the-ice-core-since-the-rectangle-is-inscribed-in-a-circle-its-diagonal-is-equal-to-the-diameter-of-the-circle-applying-the-pythagorean-theorem-to-the-rectangle-with-sides-4-cm-and-8-cm-6956252026-04-22T17:03:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-rectangular-platform-is-inscribed-in-the-circular-foundation-so-the-diagonal-of-the-rectangle-equals-the-diameter-of-the-circle-using-the-pythagorean-theorem-52529812026-04-22T17:03:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-diameter-of-the-circular-unit-is-equal-to-the-diagonal-of-the-3-cm-by-4-cm-rectangle-using-the-pythagorean-theorem-60230042026-04-22T17:03:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-m-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-s-is-36989612026-04-22T17:03:19+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-verify-that-the-triangle-is-right-angled-check-13712432026-04-22T17:03:16+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-a-be-the-set-of-bees-and-b-the-set-of-butterflies-3033452026-04-22T17:07:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-number-of-ways-to-choose-3-from-10-is-36443302026-04-22T17:07:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-radius--r--of-the-circumscribed-circle-of-a-triangle-with-sides--a--13---b--14--and--c--15--we-use-the-formula-46286862026-04-22T17:17:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-length-of-the-shortest-altitude-corresponds-to-the-longest-side-which-is-11-km-to-find-this-altitude-we-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--is-79971202026-04-22T17:17:10+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-th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-since-each-test-run-uses-128-mb-and-processes-16-profiles-the-memory-per-profile-is--6720342026-04-23T04:29:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--s--represent-the-number-of-simulations-that-can-run-simultaneously-since-each-simulation-uses-32-gb-of-ram-the-total-ram-required-is--32s--the-total-available-ram-is-512-gb-so-we-solve-86045922026-04-23T04:29:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--n--represent-the-number-of-data-entries-since-each-entry-contributes-005-seconds-the-total-time-is--005n--the-constraint-is-93481042026-04-23T04:29:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--s--represent-the-number-of-simulations-each-simulation-requires-2-cpus-and-8-gb-ram-so-the-constraints-are-81975672026-04-23T04:29:13+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-8-and-14-we-first-find-the-prime-factorization-of-each-number-40485322026-04-23T04:34:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-when-both-processes-align-we-compute-the-least-common-multiple-lcm-of-12-and-18-73861882026-04-23T04:34:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-the-asteroids-align-we-compute-the-least-common-multiple-lcm-of-15-and-25-552352026-04-23T04:34:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-reactions-align-we-compute-the-least-common-multiple-lcm-of-10-and-14-51929332026-04-23T04:34:28+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-when-both-notifications-align-we-compute-the-least-common-multiple-lcm-of-9-and-15-38192642026-04-23T04:34:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-crater-patterns-align-we-compute-the-least-common-multiple-lcm-of-6-and-9-22420392026-04-23T04:34:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-functions-execute-together-we-compute-the-least-common-multiple-lcm-of-6-and-10-88827562026-04-23T04:34:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-rationalize-the-denominator-multiply-numerator-and-denominator-by-the-conjugate-of-the-denominator-93290622026-04-23T04:34:35+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-value-of--x--that-makes-the-two-vectors-orthogonal-we-compute-their-dot-product-and-set-it-equal-to-zero-8589922026-04-24T18:37:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-us-define--fu--based-on-the-given-functional-equation-given-94267972026-04-24T18:37:40+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given--pp--q--45--and--p--q--9--86918142026-04-23T04:38:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x--be-the-number-of-successful-trials-in-the-additional-10-the-initial-success-rate-is--065-imes-40--26--successful-attempts-after-10-more-trials-total-attempts-are--50--and-total-successes-are--26--x--the-new-success-rate-is--23639122026-04-23T04:38:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--x--be-project-bs-allocation-then-project-a-receives--x--40000--the-equation-is--x--x--40000--240000--simplify--2x--200000--so--x--100000--72393402026-04-23T04:38:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-base-nine-number--2439--means-98980662026-04-23T04:38:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-we-calculate-the-total-number-of-ways-to-select-and-arrange-5-manuscripts-from-10-ensuring-that-at-least-2-are-astronomy-related-we-consider-three-valid-cases-exactly-2-astronomy-manuscripts-exactly-3-and-exactly-4-13379412026-04-23T04:38:21+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--o--be-the-cost-of-an-oxygen-tank-and--w--the-cost-of-a-water-purifier-the-system-of-equations-is-72710722026-03-15T18:14:11+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--a--be-the-legs-per-antelope-and--z--the-legs-per-zebra-given--a--4--and--z----the-total-legs-are-90118832026-03-15T18:14:12+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-from--p--q--7--express--q--7---p--substitute-into-the-first-equation-12286042026-03-15T18:14:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-herpetologist-is-observing-9-distinct-species-5-frogs--4-snakes-and-each-species-is-observed-exactly-once-over-9-days-since-all-species-are-distinct-and-order-matters-the-number-of-sequences-is-simply-the-number-of-permutations-of-9-distinct-items-59674382026-03-15T18:14:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-8-signals-where-there-are-3-ss-3-ls-and-2-ps-since-the-total-number-of-signals-is-8-and-some-are-repeated-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-46019142026-03-15T18:14:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-7-episodes-where-3-are-on-physicists-p-2-on-chemists-c-and-2-on-biologists-b-since-videos-within-each-field-are-indistinguishable-the-number-of-distinct-arrangements-is-33413992026-03-15T18:14:17+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-7-sensors-3-ts-2-ss-and-2-ds-since-sensors-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-36975722026-03-15T18:14:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-arranging-6-videos-with-2-from-each-of-3-topics-calculus-c-geometry-g-number-theory-n-the-number-of-distinct-sequences-is-219152026-03-15T18:14:18+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-original-area-using-herons-formula-42204492026-04-23T04:39:33+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-verify-that-the-triangle-is-valid-using-the-triangle-inequality-33160012026-04-24T18:39:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-simplify-the-left-hand-side-22414892026-04-23T05:04:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-expand-both-sides-of-the-equation-31201702026-04-23T05:04:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--t--10--into-the-formula-53638072026-04-23T05:05:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-velocity-is-the-derivative-of-position-38569422026-04-23T05:05:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the--t--intercept-occurs-when--d--0--64843262026-04-23T05:05:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--m--be-the-cost-of-a-lab-manual-and--g--be-the-cost-of-a-safety-goggles-the-system-of-equations-is-13043982026-04-24T18:46:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--a--be-the-cost-per-cubic-meter-of-a-and--b--for-b-the-system-is-84882922026-04-24T18:46:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--p--be-the-cost-of-a-plastic-bottle-and--s--for-a-soda-case-the-system-is-42233002026-04-24T18:46:00+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-the-groups-are-indistinguishable-and-each-must-receive-at-least-one-project-the-problem-reduces-to-partitioning-6-distinct-projects-into-4-non-empty-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s6-4--the-formula-for--sn-k--is-52335412026-04-23T05:10:05+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-3-modules-where-7-are-distinguishable-prototypes-and-5-are-identical-backup-units-since-the-prototype-units-are-unique-and-the-backup-units-are-indistinguishable-we-consider-combinations-based-on-how-many-prototypes-are-selected-from-0-to-3-and-how-many-backup-units-are-used-up-to-3-but-limited-by-availability-and-total-draw-90175212026-04-23T05:14:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-selecting-one-indistinguishable-stem-cell-line-from-4-identical-types-so-only-one-choice-for-the-stem-cell-category-and-three-distinct-neural-templates-from-6-distinguishable-ones-since-the-stem-cell-line-is-identical-across-all-selecting-any-one-gives-one-unique-type-11508322026-04-23T05:14:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since--fx--is-a-cubic-polynomial-its-third-differences-are-constant-we-compute-the-values-6013432026-04-23T05:17:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-using-the-binomial-probability-formula-with--n--6---p--03--and--k--2--18401182026-04-23T05:20:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-12-15-and-20-we-begin-by-factoring-each-number-into-its-prime-components-87311922026-04-23T05:20:23+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-system-of-congruences-48056332026-04-23T05:20:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-looking-for-the-smallest-integer-greater-than-100-that-is-a-common-multiple-of-18-and-24-but-not-a-multiple-of-30-first-compute-the-least-common-multiple-lcm-of-18-and-24-78706352026-04-23T05:20:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-when-both-cycles-align-we-compute-the-least-common-multiple-lcm-of-36-and-60-57459752026-04-23T05:20:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-compute-the-probability-that-exactly-2-out-of-5-students-receive-their-originally-assigned-experiment-a-fixed-set-of-permutations-with-the-remaining-3-elements-forming-a-derangement-with-no-fixed-points-among-the-other-students-27430342026-04-23T05:20:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-maximum-population-we-first-compute-the-derivative-of--pt--45438952026-04-23T05:24:52+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-progression-rate-is--ln--65732502026-04-23T05:24:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-substitute--h--3--into-the-function-17382882026-04-23T05:24:59+00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ting-with--p0--2000--the-recurrence-relation-is-29143762026-04-23T05:54:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-original-side-length-is--s--new-side-length-is--s---3--the-area-of-a-regular-hexagon-is--81786482026-04-23T05:56:32+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-find-the-largest-perfect-square-divisor-of-2025-37519232026-04-23T06:03:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-find-the-largest-perfect-square-that-divides-1024-82672232026-04-23T06:03:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-radius--r--of-the-inscribed-circle-in-any-triangle-is-given-by-36725752026-04-23T06:03:46+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-estimate-the-limit-we-find-the-fixed-point-of--fx--x---20426072026-04-23T06:08:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-gears-is-6--5--4--15-we-want-the-probability-that-exactly-2-of-the-3-selected-gears-are-blue-19145452026-04-23T06:08:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-rolling-an-8-sided-die-four-times-we-want-exactly-24429202026-04-23T06:08:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-total-number-of-distinct-permutations-of-a-multiset-is-given-by-dividing-the-factorial-of-the-total-number-of-letters-by-the-product-of-the-factorials-of-the-counts-of-each-repeated-letter-here-there-are-7-letters-in-total-56518882026-04-23T06:16:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-code-fixes-4-digits-d-e-f-and-permutes-only-a-b-c-in-the-remaining-3-positions-the-total-number-of-permutations-of-a-b-c-is-3--6-among-these-the-number-of-permutations-where-a-comes-before-b-is-exactly-half-since-in-half-the-arrangements-a-precedes-b-and-in-half-b-precedes-a-the-cases-are-symmetric-87376942026-04-23T06:16:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is--15--m-first-compute-the-area-using-herons-formula-the-semiperimeter--s--is-93452622026-04-23T06:16:47+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-of-the-full-sphere-of-radius--y--is-13756692026-04-23T06:16:48+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-permutation-of-a-multiset-9-total-items-with-repetitions-the-number-of-distinct-sequences-is-28009332026-04-23T06:17:20+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-five-distances-in-arithmetic-sequence-be--a---2d-a---d-a-a--d-90813002026-04-23T06:17:14+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-equation-in-cylindrical-coordinates-23175462026-04-23T06:20:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-compute-the-sum-of-the-four-numbers-31979472026-04-23T06:20:54+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-four-consecutive-odd-integers-are--11071-11073-11075-11077--their-sum-is-10687472026-04-23T06:20:56+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-four-consecutive-odd-integers-be--n-n2-n4-n6--where--n--is-odd-the-product-is-53546402026-04-23T06:21:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-convert--27538--to-base-ten-expand-using-powers-of-8-8810042026-04-23T06:21:31+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-intersection-set-the-two-equations-equal-60838332026-04-23T06:30:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-evaluate--f3--and--g3--68483152026-04-23T06:30:36+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-solve-for--x--in-the-equation--2x--3--11--68706712026-04-23T06:30:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-y-intercept-occurs-when--x--0--substitute--x--0--into-the-equation-80961202026-04-23T06:30:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-set-the-two-equations-equal-23386042026-04-23T06:30:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-distributing-6-distinguishable-fish-species-into-4-indistinguishable-zones-with-each-zone-receiving-at-least-one-species-this-is-equivalent-to-finding-the-number-of-ways-to-partition-a-set-of-6-labeled-elements-into-exactly-4-non-empty-unlabeled-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted--sn-k--here--n--6---k--4--36429942026-04-23T06:34:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-sequences-of-length-7-using-symbols-w-m-c-such-that-53949262026-04-23T06:34:29+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-hypergeometric-probability-problem-we-have-71917532026-04-23T06:34:30+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-to-count-the-number-of-ways-to-distribute-8-distinguishable-fish-into-3-distinguishable-zones-a-b-c-with-42465822026-04-23T06:34:30+00:00daily0.8https://i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circle-the-side-length-equals-the-radius-of-the-circle-64234592026-04-23T07:32:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-least-common-multiple-lcm-of-18-24-and-36-we-first-perform-prime-factorization-74199872026-04-23T07:34:37+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-determine-the-probability-we-first-count-the-positive-divisors-of-120-that-are-less-than-or-equal-to-60-the-prime-factorization-of-120-is-26319572026-04-23T07:34:38+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-factor-numerator-and-denominator-90819332026-03-15T21:15:39+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-local-minima-first-compute-the-derivative-37276502026-04-23T07:36:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-polynomial-46527482026-04-23T07:36:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-each-term-in-the-first-polynomial-by-each-term-in-the-second-34699472026-04-23T07:38:27+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-mass-of-one-astrolabe-be--m--kilograms-since-there-are-three-identical-astrolabes-their-total-mass-is--3m--this-is-given-as-27-kilograms-14048492026-04-23T07:41:50+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-arithmetic-sequence-be--a-a--d-a--2d-a--3d-a--4d--89586022026-04-23T07:43:59+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-tout-dabord-convertissons-3-livres-en-kilogrammes--57521342026-04-23T07:56:34+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-we-are-given-the-volume-formula--v--57077072026-04-23T07:56:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-five-weights-be--70872012026-04-23T07:56:41+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-find-the-largest-square-plot-size-we-compute-the-greatest-common-divisor-gcd-of-72-and-48-58715482026-04-24T19:27:26+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-an-arithmetic-sequence-with-first-term--a--3--common-difference--d--5--and-last-term--l--198--52812092026-04-23T07:59:02+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-multiply-both-sides-by--x---1x--1--to-eliminate-denominators-70728602026-04-23T08:00:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-composition--fgx--is-defined-when--gx--is-defined-and-when--gx--lies-in-the-domain-of--f--since--fx--is-a-polynomial-its-defined-for-all-real-numbers-so-we-only-need--gx--to-be-defined-89748652026-04-23T08:00:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-the-expression-under-the-square-root-must-be-non-negative-25924422026-04-23T08:00:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-given--x--y--z--1--note-that--y--z--1---x---z--x--1---y---x--y--1---z--so-the-expression-becomes-94040382026-04-23T08:00:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-to-solve-this-first-compute-each-square-and-sum-them-87851592026-04-23T08:37:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-find-the-least-common-multiple-of-12-and-18-17327842026-04-23T08:37:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-first-treat-each-species-as-a-single-block-there-are-3-ways-to-arrange-the-3-blocks-chimpanzees-gorillas-orangutans-within-each-block-the-animals-can-be-permuted-among-themselves-5-for-chimpanzees-3-for-gorillas-and-4-for-orangutans-the-total-number-of-arrangements-is-38674162026-04-23T08:37:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-this-is-a-multinomial-problem-where-we-count-the-number-of-permutations-of-a-multiset-the-total-number-of-sequences-is-8385552026-04-23T08:37:51+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let-the-two-digit-number-be--n--we-are-given-75144042026-04-23T08:40:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-let--n--be-the-two-digit-number-such-that-61391742026-04-23T08:40:03+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-since-no-two-adjacent-blocks-can-be-the-same-color-and-we-have-equal-numbers-of-red-and-blue-blocks-the-only-possible-alternating-patterns-are-rbrbrb-and-brbrbr-these-are-the-only-two-sequences-that-satisfy-the-condition-and-use-3-red-and-3-blue-blocks-no-other-arrangement-avoids-adjacent-duplicates-for-example-starting-with-b-leads-to-brbrbr-and-starting-with-r-leads-to-rbrbrb-no-other-starting-point-avoids-conflict-thus-there-are-exactly-2-valid-arrangements-44220482026-04-23T08:42:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-area--a--of-a-right-triangle-is-given-by--a--13338402026-04-23T08:42:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-volume-of-a-hemisphere-of-radius--r--is-64898912026-04-23T08:42:53+00:00daily0.8https://iqdig1.iqnection.com/resources/solution-the-semiperimeter--s--of-the-triangle-is-38878332026-04-23T08:42:53+00:00daily0.8https://iqdig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