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NMO Selection Tests for the Junior Balkan Mathematical Olympiad

Algebra Difficulty 4.6 AIME Prove it Romania

Given x1,x2,…,xnx_1, x_2, \dots, x_n real numbers, prove that there exists a real number yy such that
{y−x1}+{y−x2}+⋯+{y−xn}≤n−12. \{y - x_1\} + \{y - x_2\} + \dots + \{y - x_n\} \le \frac{n-1}{2}.

Solution

As {a}+{−a}≤1,∀a∈R\{a\} + \{-a\} \le 1, \forall a \in \mathbb{R} (with equality if aa is not an integer), we have
∑1≤i≠j≤n{xi−xj}≤n(n−1)2, \sum_{1 \le i \ne j \le n} \{x_i - x_j\} \le \frac{n(n-1)}{2},

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