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forge 1.12.2 - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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📌 Solution: To minimize the expression

📌 S = \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x},

📌 given that $ x + y + z = 1 $ and $ x, y, z > 0 $, we apply the **Cauchy-Schwarz Inequality** in the following form:

📌 Since $ x + y + z = 1 $, we substitute into the inequality:

📌 To check if this bound is achievable, consider the equality condition in Cauchy-Schwarz: equality occurs when

📌 Let this common value be $ k $. Then,

📌 \frac{x^2}{y} = \frac{y^2}{z} \Rightarrow x^2 z = y^3, \quad \frac{y^2}{z} = \frac{z^2}{x} \Rightarrow y^2 x = z^3.

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