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what started the war of 1812 - ECD Germany

📅 May 8, 2026 👤 scraface

May 08, 2026

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📌 Question: Let $ p $ and $ q $ be real numbers such that $ p + q = 5 $ and $ pq = 6 $. Find the value of $ p^3 + q^3 $.

📌 Solution: We are given $ p + q = 5 $ and $ pq = 6 $. We use the identity:

📌 p^3 + q^3 = (p + q)^3 - 3pq(p + q)

📌 p^3 + q^3 = 5^3 - 3 \cdot 6 \cdot 5 = 125 - 90 = 35

📌 Question: Let $ a $, $ b $, and $ c $ be positive real numbers such that $ a + b + c = 1 $. Find the minimum value of $ rac{1}{a} + rac{1}{b} + rac{1}{c} $.

📌 Solution: We are to minimize $ rac{1}{a} + rac{1}{b} + rac{1}{c} $ under the constraint $ a + b + c = 1 $ with $ a, b, c > 0 $.

📌 By the AM-HM inequality:

📌 rac{a + b + c}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}} \Rightarrow rac{1}{3} \geq rac{3}{rac{1}{a} + rac{1}{b} + rac{1}{c}}

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