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Home » 19y^2 \equiv 0 \pmod25 \Rightarrow y^2 \equiv 0 \pmod25 \Rightarrow y \equiv 0 \pmod5. - Crosslake

19y^2 \equiv 0 \pmod25 \Rightarrow y^2 \equiv 0 \pmod25 \Rightarrow y \equiv 0 \pmod5. - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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.

📌 From these, we deduce a symmetry: suppose $ x = y = z $. Then $ x + y + z = 3x = 1 \Rightarrow x = \frac{1}{3} $. Substituting into $ S $:

📌 S = 3 \cdot \frac{(1/3)^2}{1/3} = 3 \cdot \frac{1/9}{1/3} = 3 \cdot \frac{1}{3} = 1.

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