HMMT 十一月 2020 · 冲刺赛 · 第 23 题

HMMT November 2020 — Guts Round — Problem 23

[12] Two points are chosen inside the square { ( x, y ) | 0 ≤ x, y ≤ 1 } uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area a of the union of the two squares can be expressed as , where a, b are relatively prime positive integers. b Compute 100 a + b .

解析

[12] Two points are chosen inside the square { ( x, y ) | 0 ≤ x, y ≤ 1 } uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area a of the union of the two squares can be expressed as , where a, b are relatively prime positive integers. b Compute 100 a + b . Proposed by: Yannick Yao Answer: 1409 Solution: B 1 − ∆ y A 1 − ∆ x Let ∆ x and ∆ y be the positive differences between the x coordinates and y coordinates of the centers of the squares, respectively. Then, the length of the intersection of the squares along the x dimension is 1 − ∆ x , and likewise the length along the y dimension is 1 − ∆ y . In order to find the expectation of ∆ x and ∆ y , we can find the volume of the set of points ( a, b, c ) such that 0 ≤ a, b ≤ 1 and c ≤ | a − b | . 1 This set is composed of the two pyramids of volume shown below: 6 1 Since the expected distance between two points on a unit interval is therefore , we have that 3 2 E [1 − ∆ x ] = E [1 − ∆ y ] = . The expectation of the product of independent variables equals the 3 4 product of their expectations, so the expected area of intersection is and the expected area of union 9 4 14 is 2 − = . 9 9