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Part of 2017 Poland - Second Round

Problems(1)

Inequality on 2n-1 real numbers [Poland Second Round 2017, D1, P3]

Source: Polish Mathematical Olympiad Second Round, Day 1, Problem 3

2/25/2017
Let x1≤x2≤…≤x2n−1x_1 \le x_2 \le \ldots \le x_{2n-1} be real numbers whose arithmetic mean equals AA. Prove that 2∑i=12n−1(xi−A)2≥∑i=12n−1(xi−xn)2.2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.
inequalitiesn-variable inequality