Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Find the answer

Let $A_{1}, A_{2}, \ldots, A_{2015}$ be distinct points on the unit circle with center $O$ . For every two distinct integers $i, j$ , let $P_{i j}$ be the midpoint of $A_{i}$ and $A_{j}$ . Find the smallest possible value of $\sum_{1 \leq i<j \leq 2015} O P_{i j}^{2}$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Use vectors. $\sum\left|a_{i}+a_{j}\right|^{2} / 4=\sum\left(2+2 a_{i} \cdot a_{j}\right) / 4=\frac{1}{2}\binom{2015}{2}+\frac{1}{4}\left(\left|\sum a_{i}\right|^{2}-\sum\left|a_{i}\right|^{2}\right) \geq 2015 \cdot \frac{2014}{4}-\frac{2015}{4}=\frac{2015 \cdot 2013}{4}$ , with equality if and only if $\sum a_{i}=0$ , which occurs for instance for a regular 2015-gon.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.