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📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Let $AB = BC = CD = DA = DB = DC = s$. Then we solve the system of equations:

📌 Subtracting the first equation from the second and the second from the third gives linear equations in $x, y, z$. Subtracting the first from the second:

📌 [(x - 4)^2 - (x - 1)^2] + [(y - 5)^2 - (y - 2)^2] + [(z - 6)^2 - (z - 3)^2] = 0

📌 -6x -6y -6z + 63 = 0 \Rightarrow x + y + z = 10.5

📌 This is not possible with integer coordinates. Hence, no such integer-coordinate point $D$ exists.

📌 Therefore, the answer is:

📌 \boxed{\text{No such point with integer coordinates exists.}}

📌 Question: A patent attorney is reviewing a software algorithm that maps user behavior on a 2D grid, where each user’s activity is represented as a vector $\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$. For the system to ensure fairness, the algorithm must always preserve orthogonality: if $\mathbf{v} \cdot \begin{pmatrix} 3 \\ -4 \end{pmatrix} = 0$, then it must also imply $\mathbf{v} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = 0$ for some integer vector $\begin{pmatrix} x \\ y \end{pmatrix}$. What must be true about all such vectors $\mathbf{v}$?

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