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Home » إذن، $u = \frac12$ أو $u = -1$. - Londonproperty

إذن، $u = \frac12$ أو $u = -1$. - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(y + z + x) \geq (x + y + z)^2

📌 Since $ x + y + z = 1 $, we have:

📌 \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(1) \geq 1^2 = 1

📌 \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \geq 1

📌 Equality holds when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we get $ x = y = z = \frac{1}{3} $.

📌 Check the expression at $ x = y = z = \frac{1}{3} $:

📌 \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} = 3 \cdot \frac{(1/3)^2}{1/3} = 3 \cdot \frac{1/9}{1/3} = 3 \cdot \frac{1}{3} = 1

📌 Question: For all real numbers $ a, b, c $, find the number of functions $ f : \mathbb{R} \to \mathbb{R} $ such that

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