HMMT 二月 2025 · 冲刺赛 · 第 8 题

HMMT February 2025 — Guts Round — Problem 8

[6] A checkerboard is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are distinct if and only if they have a different number of rows or columns. For example, a 20 × 25 checkerboard and a 25 × 20 checkerboard are considered distinct. Compute the number of distinct checkerboards that have exactly 41 black cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2025, February 15, 2025 — GUTS ROUND Organization Team Team ID#

解析

[6] A checkerboard is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are distinct if and only if they have a different number of rows or columns. For example, a 20 × 25 checkerboard and a 25 × 20 checkerboard are considered distinct. Compute the number of distinct checkerboards that have exactly 41 black cells. Proposed by: Albert Wang Answer: 9 Solution: Since there is a black corner on the checkerboard, the number of white squares is at most the number of black squares. So, the board either has 40 or 41 white squares. Therefore, we want to 4 compute the number of ordered pairs ( r, c ) with a product of 81 or 82. Since 81 = 3 has 5 divisors and 82 = 41 · 2 has 4 divisors, there are 9 checkerboards with exactly 41 black cells.