HMMT 二月 2003 · 冲刺赛 · 第 23 题
HMMT February 2003 — Guts Round — Problem 23
题目详情
- [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
- 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.