HMMT 十一月 2022 · 冲刺赛 · 第 13 题

HMMT November 2022 — Guts Round — Problem 13

[9] Consider the paths from (0 , 0) to (6 , 3) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the x -axis, and the line x = 6 over all such paths. (In particular, the path from (0 , 0) to (6 , 0) to (6 , 3) corresponds to an area of 0 . )

解析

[9] Consider the paths from (0 , 0) to (6 , 3) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the x -axis, and the line x = 6 over all such paths. (In particular, the path from (0 , 0) to (6 , 0) to (6 , 3) corresponds to an area of 0 . ) Proposed by: Andrew Lee Answer: 756 Solution: We see that the sum of the areas under the path is equal the sum of the areas above the path. Thus, the sum of the areas under the path is half the area of the rectangle times the number of 9 18 ( ) 3 paths, which is = 756. 2