HMMT 二月 2005 · 冲刺赛 · 第 15 题

HMMT February 2005 — Guts Round — Problem 15

[7] Let S be the set of lattice points inside the circle x + y = 11. Let M be the greatest area of any triangle with vertices in S . How many triangles with vertices in S have area M ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 19, 2005 — GUTS ROUND

解析

Let S be the set of lattice points inside the circle x + y = 11. Let M be the greatest area of any triangle with vertices in S . How many triangles with vertices in S have area M ? Solution: 16 The boundary of the convex hull of S consists of points with ( x, y ) or ( y, x ) = (0 , ± 3), ( ± 1 , ± 3), and ( ± 2 , ± 2). For any triangle T with vertices in S , we can increase its area by moving a vertex not on the boundary to some point on the boundary. Thus, 5 if T has area M , its vertices are all on the boundary of S . The next step is to see (either by inspection or by noting that T has area no larger than that of an equilateral √ triangle inscribed in a circle of radius 10, which has area less than 13) that M = 12. There are 16 triangles with area 12, all congruent to one of the following three: vertices (2 , 2), (1 , − 3), and ( − 3 , 1); vertices (3 , − 1), ( − 3 , − 1), and (1 , 3); or vertices (3 , − 1), ( − 3 , − 1), and (0 , 3).