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Home » From Shadow Operatives to Blind Spy Missions – The Ultimate Stealth Movie Ranking! - Dyverse

From Shadow Operatives to Blind Spy Missions – The Ultimate Stealth Movie Ranking! - Dyverse

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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

📌 Solution: Consider the general term:

📌 \frac{1}{\sqrt{n} + \sqrt{n+1}}

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A tank can be filled by Pipe A in 4 hours and by Pipe B in 6 hours. If...
Rate of Pipe A = \( \frac{1}{4} \) of the tank per hour
Rate of Pipe B = \( \frac{1}{6} \) of the tank per hour
When both are open for 2 hours, they fill:
\[ 2 \left( \frac{1}{4} + \frac{1}{6} \right) = 2 \left( \frac{3}{12}...
Remaining tank to fill = \( 1 - \frac{5}{6} = \frac{1}{6} \)
Now, only Pipe A is open, filling at \( \frac{1}{4} \) per hour. Time...
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