HMMT 二月 2022 · 冲刺赛 · 第 27 题

HMMT February 2022 — Guts Round — Problem 27

[14] In three-dimensional space, let S be the region of points ( x, y, z ) satisfying − 1 ≤ z ≤ 1. Let S , S , . . . , S be 2022 independent random rotations of S about the origin (0 , 0 , 0). The expected 1 2 2022 aπ volume of the region S ∩ S ∩ · · · ∩ S can be expressed as , for relatively prime positive integers a 1 2 2022 b and b . Compute 100 a + b .

解析

[14] In three-dimensional space, let S be the region of points ( x, y, z ) satisfying − 1 ≤ z ≤ 1. Let S , S , . . . , S be 2022 independent random rotations of S about the origin (0 , 0 , 0). The expected 1 2 2022 aπ volume of the region S ∩ S ∩ · · · ∩ S can be expressed as , for relatively prime positive integers 1 2 2022 b a and b . Compute 100 a + b . Proposed by: Daniel Zhu Answer: 271619 Solution: Consider a point P of distance r from the origin. The distance from the origin of a random projection of P onto a line is uniform from 0 to r . Therefore, if r < 1 then the probability of P being − 2022 in all the sets is 1, while for r ≥ 1 it is r . Therefore the volume is Z ∞ 4 π 1 1 2696 π 2 − 2022