HMMT 二月 2008 · 冲刺赛 · 第 6 题

英文解析

6. [ 6 ] Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines ofthe following figure.
   Answer: 297 First, let us count the total number of rectangles in the grid without the hole in the 1
   ( )
   middle. There are = 21 ways to choose the two vertical boundaries of the rectangle, and there are 7
   21 ways to choose the two horizontal boundaries of the rectangles. This makes 21 = 441 rectangles.22
   However, we must exclude those rectangles whose boundary passes through the center point. We cancount these rectangles as follows: the number of rectangles with the center of the grid lying in theinterior 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 overall 4 orientations, we see that the total number of rectangles to exclude is 4(27 + 9) = 144. Therefore,
   the answer is 441 − 144 = 297.
   11 HARVARD-MIT MATHEMATICS TOURNAMENT, 23 FEBRUARY 2008 — GUTS ROUNDth