HMMT 二月 2008 · 冲刺赛 · 第 6 题

HMMT February 2008 — Guts Round — Problem 6

[ 6 ] Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND

解析

[ 6 ] Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure. 1 Answer: 297 First, let us count the total number of rectangles in the grid without the hole in the ( ) 7 middle. There are = 21 ways to choose the two vertical boundaries of the rectangle, and there are 2 2 21 ways to choose the two horizontal boundaries of the rectangles. This makes 21 = 441 rectangles. However, we must exclude those rectangles whose boundary passes through the center point. We can count these rectangles as follows: the number of rectangles with the center of the grid lying in the interior of its south edge is 3 × 3 × 3 = 27 (there are three choices for each of the three other edges); the number of rectangles whose south-west vertex coincides with the center is 3 × 3 = 9. Summing over all 4 orientations, we see that the total number of rectangles to exclude is 4(27 + 9) = 144. Therefore, the answer is 441 − 144 = 297. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUND