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Home » 100^2 = 5200 + 2xy - Dyverse

100^2 = 5200 + 2xy - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}

📌 This is a well-known inequality in symmetric form. By the **Nesbitt’s Inequality**, for positive real numbers \( a, b, c \),

📌 \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} \geq \frac{3}{2}

📌 with equality if and only if \( a = b = c \). Since \( a + b + c = 1 \), equality occurs when \( a = b = c = \frac{1}{3} \). Substituting, each term becomes:

📌 \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}

📌 So the sum is \( 3 \cdot \frac{1}{2} = \frac{3}{2} \).

📌 Thus, the minimum value is \( \boxed{\frac{3}{2}} \).

📌 Question: Compute \( \sum_{n=1}^{50} \frac{1}{\sqrt{n} + \sqrt{n+1}} \).

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