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Prove this inequality

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020
inequalitiesSummation

Problem Statement

Let x1,x2,≥,xnx_1, x_2,\geq , x_n be a positive numbers, k≥1k \geq 1. Then the following inequality is true: (x1k+x2k+⋯+xnk)k+1≥(x1k+1+x2k+1⋯+xnk+1)k+2(∑1≤i<j≤nxikxj)k\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k
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