HMMT 二月 2022 · 冲刺赛 · 第 24 题

HMMT February 2022 — Guts Round — Problem 24

[12] Let S be a unit square in the Cartesian plane with horizontal and vertical sides. For any n > 0, the 0 shape S is formed by adjoining 9 copies of S in a 3 × 3 grid, and then removing the center copy. For n n − 1 example, S is shown below: 3 ′ ′ ′ ′ Let a be the expected value of | x − x | + | y − y | , where ( x, y ) and ( x , y ) are two points chosen randomly n within S . There exist relatively prime positive integers a and b such that n a a n lim = . n n →∞ 3 b Compute 100 a + b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2022, February 19, 2022 — GUTS ROUND Organization Team Team ID#

解析

[12] Let S be a unit square in the Cartesian plane with horizontal and vertical sides. For any n > 0, 0 the shape S is formed by adjoining 9 copies of S in a 3 × 3 grid, and then removing the center n n − 1 copy. For example, S is shown below: 3 ′ ′ ′ ′ Let a be the expected value of | x − x | + | y − y | , where ( x, y ) and ( x , y ) are two points chosen n randomly within S . There exist relatively prime positive integers a and b such that n a a n lim = . n n →∞ 3 b Compute 100 a + b . Proposed by: Gabriel Wu Answer: 1217 Solution: By symmetry, we only need to consider the x -distance, then we can multiply our answer by