HMMT 二月 2003 · 冲刺赛 · 第 23 题
HMMT February 2003 — Guts Round — Problem 23
题目详情
英文原题
- [9] How many lattice points are enclosed by the triangle with vertices (0 , 99), (5 , 100),
and (2003 , 500)? Don’t count boundary points.
解析
英文解析
- How many lattice points are enclosed by the triangle with vertices (0 , 99), (5 , 100), and
(2003 , 500)? Don’t count boundary points.
Solution: 0
Using the determinant formula, we get that the area of the triangle is
∣ ∣
∣ ∣ /
5 1
∣ ∣
∣ ∣ 2 = 1 .
∣ ∣
2003 401
There are 4 lattice points on the boundary of the triangle (the three vertices and
(1004 , 300)), so it follows from Pick’s Theorem that there are 0 in the interior.