HMMT 二月 2008 · TEAM1 赛 · 第 5 题

英文解析

5. [ 10 ] Let S be a set of 5 points in the 2-dimensional lattice. Show that we can always choosea 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 musthave the same parity in both coordinates (i.e., they are congruent in mod 2). Then, themidpoint of these two points must be a lattice point.
   d 2