HMMT 十一月 2025 · 冲刺赛 · 第 5 题

HMMT November 2025 — Guts Round — Problem 5

[6] A circle of radius strictly less than 2 is drawn in the plane. Compute the maximum possible number of lattice points that can lie on its circumference. (A lattice point is a point with integer coordinates.)

解析

[6] A circle of radius strictly less than 2 is drawn in the plane. Compute the maximum possible number of lattice points that can lie on its circumference. (A lattice point is a point with integer coordinates.) Proposed by: Jordan Lefkowitz Answer: 8 Solution: Observe that every lattice point must lie on a horizontal grid line (i.e., a line y = k where k is an integer). Then note the following: • The circle must intersect at most 4 horizontal grid lines. This is because for any set of 5 horizontal grid lines, the distance between the topmost and bottommost lines is at least 4. But the maximum distance between any two points on the circle is its diameter, which is less than 4, so it cannot intersect all 5 lines. • Each horizontal grid line can lead to at most 2 lattice point intersections, since any circle intersects a line at most twice. Thus, the number of lattice point intersections is at most 4 · 2 = 8 . For a construction, consider the 2 2 1 1 5 circle x − + y − = , which intersects 8 lattice points as shown below. 2 2 2