HMMT 二月 2008 · TEAM1 赛 · 第 5 题

HMMT February 2008 — TEAM1 Round — Problem 5

[ 10 ] Let S be a set of 5 points in the 2-dimensional lattice. Show that we can always choose a pair of points in S whose midpoint is also a lattice point. d

解析

[ 10 ] Let S be a set of 5 points in the 2-dimensional lattice. Show that we can always choose a pair of points in S whose midpoint is also a lattice point. Solution: Consider the parities of the coordinates. There are four possibilities: (odd, odd), (odd, even), (even, odd), (even, even). By the pigeonhole principle, two of the points must have the same parity in both coordinates (i.e., they are congruent in mod 2). Then, the midpoint of these two points must be a lattice point. 2 d