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Home » $ 12 = 2^2 \cdot 3 $ - Silent Sales Machine

$ 12 = 2^2 \cdot 3 $ - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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We are to minimize
a + b + c - \log\left(rac{x}{z} + rac{z}{x}
ight) = a + b + c - \log\left(rac{1}{b} + b
since $ rac{x}{z} = rac{x}{y} \cdot rac{y}{z} = ac = rac{1}{b} $,...
a + b + c - \log\left(b + rac{1}{b}
Since $ abc = 1 $, we can assume $ a = rac{x}{y} $, $ b = rac{y}{z}...
a + b + c \geq 3\sqrt[3]{abc} = 3.
Equality occurs when $ a = b = c = 1 $, i.e., when $ x = y = z $.
Now consider the second term:
\log\left(b + rac{1}{b}

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