HMMT 二月 2002 · GEN1 赛 · 第 1 题
HMMT February 2002 — GEN1 Round — Problem 1
题目详情
- 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?
解析
- 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 other hand, no two points within the circle can be at a mutual distance of 2 or greater. If there are 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 same class. Since they are distinct, this means either their first or second coordinates must differ by at least 2, so their distance is at least 2, a contradiction.