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Home » Limits: u = 1 to u = 2 - Red Crowns

Limits: u = 1 to u = 2 - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 3 $. Find the minimum value of

📌 \frac{x^2}{y + z} + \frac{y^2}{z + x} + \frac{z^2}{x + y}.

📌 Solution: We are to minimize

📌 under the constraint $ x + y + z = 3 $.

📌 Note that $ y + z = 3 - x $, $ z + x = 3 - y $, $ x + y = 3 - z $. So the expression becomes:

📌 \sum \frac{x^2}{3 - x}

📌 We apply the **Cauchy-Schwarz Inequality** in the following form:

📌 \sum \frac{x^2}{

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