PUMaC 2019 · 几何（B 组） · 第 4 题

PUMaC 2019 — Geometry (Division B) — Problem 4

Suppose we choose two real numbers x, y ∈ [0 , 1] uniformly at random. Let p be the probability that the circle with center ( x, y ) and radius | x − y | lies entirely within the unit square [0 , 1] × m [0 , 1]. Then p can be written in the form , where m and n are relatively prime nonnegative n 2 2 integers. Compute m + n .

解析

Suppose we choose two real numbers x, y ∈ [0 , 1] uniformly at random. Let p be the probability that the circle with center ( x, y ) and radius | x − y | lies entirely within the unit square [0 , 1] × m [0 , 1]. Then p can be written in the form , where m and n are relatively prime nonnegative n 2 2 integers. Compute m + n . Proposed by Sam Mathers. Answer: 10 Solution: The key observation here is that ∆ BDC and ∆ BHC are in fact congruent. Then, AB × AC AB × AC = = 5 . HB × HC DB × DC First, suppose x > y , then we have the conditions x − y < y and x − y < 1 − x . The point 2 1 of intersection of these two inequalities is when x = 2 y and y = 2 x − 1 so x = and y = . 3 3 2 1 Thus, the acceptable region for ( x, y ) is within the triangle with vertices (0 , 0), ( , ), and 3 3 1 (1 , 1). This has area . Multiplying this by two since we also have the case y > x , we get an 6 1 area of so the answer is 10. 3 1