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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: Let $ a $ and $ b $ be complex numbers such that $ a + b = 4 $ and $ a^2 + b^2 = 10 $. Find $ a^3 + b^3 $.

📌 We know $ a + b = 4 $, and

📌 a^2 + b^2 = (a + b)^2 - 2ab = 16 - 2ab = 10 \Rightarrow 2ab = 6 \Rightarrow ab = 3

📌 a^3 + b^3 = 4^3 - 3 \cdot 3 \cdot 4 = 64 - 36 = \boxed{28}

📌 Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 1 $. Find the minimum value of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $.

📌 Solution: We apply the **Cauchy-Schwarz inequality** in the Engel form (also known as Titu’s Lemma):

📌 \frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \frac{a_3^2}{b_3} \ge \frac{(a_1 + a_2 + a_3)^2}{b_1 + b_2 + b_3}

📌 But our expression is $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} = \frac{1^2}{x} + \frac{2^2}{y} + \frac{3^2}{z} $

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