HMMT 二月 2022 · 冲刺赛 · 第 21 题
HMMT February 2022 — Guts Round — Problem 21
题目详情
- [12] In the Cartesian plane, let A = (0 , 0), B = (200 , 100), and C = (30 , 330). Compute the number of 1 1 ordered pairs ( x, y ) of integers so that ( x + , y + ) is in the interior of triangle ABC . 2 2 2
解析
- [12] In the Cartesian plane, let A = (0 , 0), B = (200 , 100), and C = (30 , 330). Compute the number 1 1 of ordered pairs ( x, y ) of integers so that ( x + , y + ) is in the interior of triangle ABC . 2 2 Proposed by: Ankit Bisain Answer: 31480 Solution: We use Pick’s Theorem, which states that in a lattice polygon with I lattice points in its interior and B lattice points on its boundary, the area is I + B/ 2 − 1. Also, call a point center if it is 1 1 of the form ( x + , y + ) for integers x and y . 2 2 ◦ The key observation is the following – suppose we draw in the center points, rotate 45 degrees about √ the origin and scale up by 2. Then, the area of the triangle goes to 2 K , and the set of old lattice points and center points becomes a lattice. Hence, we can also apply Pick’s theorem to this new lattice. Let the area of the original triangle be K , let I and B be the number of interior lattice points and 1 1 boundary lattice points, respectively. Let I and B be the number of interior and boundary points c c that are center points in the original triangle. Finally, let I and B be the number of interior and 2 2 boundary points that are either lattice points or center points in the new triangle. By Pick’s Theorem on both lattices, K = I + B / 2 − 1 1 1 2 K = I + B / 2 − 1 2 2 B − B 1 2 = ⇒ ( I − I ) = K − 2 1 2 B c = ⇒ I = K − . c 2 One can compute that the area is 31500. The number of center points that lie on on AB , BC , and 0+10+30 CA are 0, 10, and 30, respectively. Thus, the final answer is 31500 − = 31480. 2 2