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2013 China Western Mathematical Olympiad ,Problem 2

Source: China Lanzhou 17 Aug 2013

August 17, 2013
inductioninequalities proposedinequalities

Problem Statement

Let the integer n≥2n \ge 2, and the real numbers x1,x2,⋯ ,xn∈[0,1]x_1,x_2,\cdots,x_n\in \left[0,1\right] .Prove that∑1≤k<j≤nkxkxj≤n−13∑k=1nkxk.\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.
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