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6

Part of 2019 China Western Mathematical Olympiad

Problems(1)

2019 China Western Mathematical Olympiad Q6

Source: China Western Mathematical Olympiad 2019 Day 2 P2

8/14/2019
Let a1,a2,⋯ ,an(n≥2)a_1,a_2,\cdots,a_n (n\ge 2) be positive numbers such that a1≤a2≤⋯≤an.a_1\leq a_2 \leq \cdots \leq a_n . Prove that ∑1≤i<j≤n(ai+aj)2(1i2+1j2)≥4(n−1)∑i=1nai2i2.\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.
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