HMMT 二月 2015 · 冲刺赛 · 第 22 题

HMMT February 2015 — Guts Round — Problem 22

[ 14 ] Let A , A , . . . , A be distinct points on the unit circle with center O . For every two distinct 1 2 2015 integers i, j , let P be the midpoint of A and A . Find the smallest possible value of ij i j ∑ 2 OP . ij 1 ≤ i<j ≤ 2015

解析

[ 14 ] Let A , A , . . . , A be distinct points on the unit circle with center O . For every two distinct 1 2 2015 integers i, j , let P be the midpoint of A and A . Find the smallest possible value of ij i j ∑ 2 OP . ij 1 ≤ i<j ≤ 2015 ( ) ∑ ∑ 2015 2015 · 2013 4056195 2 1 Answer: OR Use vectors. | a + a | / 4 = (2 + 2 a · a ) / 4 = + i j i j 4 4 2 2 ∑ ∑ ∑ 1 2014 2015 2015 · 2013 2 2 ( | a | − | a | ) ≥ 2015 · − = , with equality if and only if a = 0, which i i i 4 4 4 4 occurs for instance for a regular 2015-gon. k k 2 2 − 2 The identity still holds even if z − z = 0 for some k ≥ 1 used in the telescoping argument: why? ∏ 10 3 k This indeed works, since cos(2 θ ) 6 = 0: why? k =0 Guts