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📅 May 18, 2026 👤 scraface

May 18, 2026

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5Question: Let \( a, b, c \) be positive real numbers such that \( a +...
Solution: We are given \( a + b + c = 1 \) with \( a, b, c > 0 \), so...
\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...
\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 \...
\frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}}...
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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