HMMT 二月 2002 · GEN1 赛 · 第 1 题

英文解析

1. What is the maximum number of lattice points (i.e. points with integer coordinates)
   in the plane that can be contained strictly inside a circle of radius 1?
   Solution: 4 . The circle centered at (1 / 2 , 1 / 2) shows that 4 is achievable. On the otherhand, no two points within the circle can be at a mutual distance of 2 or greater. If thereare more than four lattice points, classify all such points by the parity of their coordinates:
   (even, even), (even, odd), (odd, even), or (odd, odd). Then some two points lie in the sameclass. Since they are distinct, this means either their first or second coordinates must differby at least 2, so their distance is at least 2, a contradiction.