HMMT 十一月 2020 · 冲刺赛 · 第 14 题

HMMT November 2020 — Guts Round — Problem 14

[9] A point ( x, y ) is selected uniformly at random from the unit square S = { ( x, y ) | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } . a If the probability that (3 x +2 y, x +4 y ) is in S is , where a, b are relatively prime positive integers, compute b 100 a + b . 2 2

解析

[9] A point ( x, y ) is selected uniformly at random from the unit square S = { ( x, y ) | 0 ≤ x ≤ 1 , 0 ≤ a y ≤ 1 } . If the probability that (3 x + 2 y, x + 4 y ) is in S is , where a, b are relatively prime positive b integers, compute 100 a + b . Proposed by: Christopher Xu Answer: 820 Solution: Under the transformation ( x, y ) 7 → (3 x + 2 y, x + 4 y ), S is mapped to a parallelogram with vertices (0 , 0), (3 , 1), (5 , 5), and (2 , 4). Using the shoelace formula, the area of this parallelogram is 10. ( ) 1 The intersection of the image parallelogram and S is the quadrilateral with vertices (0 , 0), 1 , , (1 , 1), 3 ( ) 1 1 and , 1 . To get this quadrilateral, we take away a right triangle with legs 1 and and a right triangle 2 2 1 1 1 1 1 7 with legs 1 and from the unit square. So the quadrilateral has area 1 − · − · = . Then 3 2 2 2 3 12 7 7 12 the fraction of the image parallelogram that lies within S is = , which is the probability that a 10 120 point stays in S after the mapping. 2 2