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\(sum x_i )^2 <= n sum_ x_i x_{n-i+1}

Source: Czech and Slovak Olympiad 1979, National Round, Problem 4

September 13, 2024
algebrainequalities

Problem Statement

Let nn be any natural number. Find all nn-tuples of real numbers x1≤x2≤...≤xnx_1\le x_2\le ... \le x_n, for which holds (∑i=1nxi)2≤n∑i=1nxixn−i+1.\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.
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