HMMT 十一月 2019 · 冲刺赛 · 第 5 题

HMMT November 2019 — Guts Round — Problem 5

[6] A point P is chosen uniformly at random inside a square of side length 2. If P , P , P , and P are the 1 2 3 4 reflections of P over each of the four sides of the square, find the expected value of the area of quadrilateral P P P P . 1 2 3 4

解析

[6] A point P is chosen uniformly at random inside a square of side length 2. If P , P , P , and P 1 2 3 4 are the reflections of P over each of the four sides of the square, find the expected value of the area of quadrilateral P P P P . 1 2 3 4 Proposed by: Carl Schildkraut Answer: 8 Let ABCD denote the square defined in the problem. We see that if P is the reflection of P over AB , 1 then the area of P AB is the same as the area of P AB . Furthermore, if P is the reflection of P over 1 4 DA , P , A , and P are collinear, as A is the midpoint of P P . Repeating this argument for the other 1 4 1 4 points gives us that the desired area is [ P AB ]+[ P BC ]+[ P CD ]+[ P DA ]+[ ABCD ] = [ P AB ]+[ P BC ]+[ P CD ]+[ P DA ]+[ ABCD ] = 2[ ABCD ] = 8 . 1 2 3 4