AMO-Bench 2025 Problem 24

Answer

Let $n$ be an integer with $n>2$ . Real numbers $a_1, a_2, \ldots, a_n$ satisfy $\sum_{k=1}^{n} a_k = 2n, \sum_{k=1}^{n} k\,|a_k| = 4n.$ Find the minimum value of $a_1^2 + a_2^2 + \cdots + a_n^2$ .

Dataset text AMO-Bench. Open licence.

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For $n=3$ , the minimum of $a_1^2 + a_2^2 + a_3^2$ is $12$ . For $n \ge 4$ , the minimum of $a_1^2 + a_2^2 + \cdots + a_n^2$ is $\dfrac{6n^2}{5}$ .

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