HMMT 二月 1998 · 几何 · 第 10 题
HMMT February 1998 — Geometry — Problem 10
题目详情
英文原题
Question Ten [8 points].
Lukas is playing pool on a table shaped like an equilateral triangle.
The pockets are at the corners of the triangle and are labeled C , H ,
and T . Each side of the table is 16 feet long. Lukas shoots a ballfrom corner C of the table in such a way that on the second bounce,
the ball hits 2 feet away from him along side CH .
a. [5 points]
How many times will the ball bounce before hitting a pocket?
b. [2 points]
Which pocket will the ball hit?
c. [1 points]
How far will the ball travel before hitting the pocket?
解析
英文解析
- Suppose that when the ball hits a side of the table, instead of reflecting the ball’s path, wereflect the entire table over this side so that the path remains straight. If we repeatedlyreflect the table over its sides in all possible ways, we get a triangular grid that tiles the plane. Whenever the path crosses n lines in this grid parallel to CT , it will cross n lines 7
parallel to CH and n lines parallel to HT . After crosssing 8 + 7 + 15 = 30 grid lines it will 158
have crossed three lines simultaneously again, which means that the ball will have landed in 8
a pocket after bouncing 27 times. By picturing the grid it is easy to see that the pocketin question is H . The distance the ball travels during the of its trip described in the 1
°8
problem is the third side of a triangle with an 120 angle between two sides 16 and 14, which
√
2 2 °
is 16 + 14 − 2 · 16 · 14 cos 120 = 26, so length of the entire trip is 208 . 1