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Home » \(-a - 2b = -4\). - All Square Golf

\(-a - 2b = -4\). - All Square Golf

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Question: A regular tetrahedron has three of its vertices at $A = (1, 1, 1)$, $B = (1, -1, -1)$, and $C = (-1, 1, -1)$. Find the integer coordinates of the fourth vertex $D$, assuming all coordinates are integers.

📌 Solution: In a regular tetrahedron, all edges are equal. Compute distances:

📌 AB = \sqrt{(1 - 1)^2 + (1 + 1)^2 + (1 + 1)^2} = \sqrt{0 + 4 + 4} = \sqrt{8} \

📌 AC = \sqrt{(1 + 1)^2 + (1 - 1)^2 + (1 + 1)^2} = \sqrt{4 + 0 + 4} = \sqrt{8} \

📌 BC = \sqrt{(1 + 1)^2 + (-1 - 1)^2 + (-Question: Let $ x $, $ y $, and $ z $ be positive real numbers. Find the minimum value of

📌 rac{x}{y} + rac{y}{z} + rac{z}{x} - \log\left(rac{x}{z} + rac{z}{x}

📌 Let us define $ a = rac{x}{y} $, $ b = rac{y}{z} $, and $ c = rac{z}{x} $. Then, multiplying these three gives

📌 abc = rac{x}{y} \cdot rac{y}{z} \cdot rac{z}{x} = 1.

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