HMMT 二月 2021 · 冲刺赛 · 第 4 题

HMMT February 2021 — Guts Round — Problem 4

[8] In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path p , let A denote the number of p unit squares under the path p . Compute the sum of A over all up-right paths p . p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT Spring 2021, March 06, 2021 — GUTS ROUND Organization Team Team ID#

解析

[8] In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path p , let A denote the p number of unit squares under the path p . Compute the sum of A over all up-right paths p . p Proposed by: Freddie Zhao Answer: 90 ( ) 6 Solution: Each path consists of 3 steps up and 3 steps to the right, so there are = 20 total paths. 3 Consider the sum of the areas of the regions above all of these paths. By symmetry, this is the same as the answer to the problem. For any path, the sum of the areas of the regions above and below it is 2 3 = 9, so the sum of the areas of the regions above and below all paths is 9 · 20 = 180. Therefore, our 1 final answer is · 180 = 90. 2