HMMT 二月 2000 · 几何 · 第 10 题
HMMT February 2000 — Geometry — Problem 10
题目详情
英文原题
- Let C and C be t w o on en tri re e tiv e hollo w metal spheres of radius R and R 3
1 2
resp e tiv ely . F rom a p oin t P on the surfa e of C , a ra y of ligh t is emitted in w ard at 30Æfrom the radial dire tion. The ra y ev en tually returns to P . Ho w man y total re e tions 2
o of C and C do es it tak e?
1 2
解析
英文解析
- Consider the attened v ersion of the situation. Then let O be the en ter fo the spheres,
A be the rst re e tion p oin t, B be the p oin t of C such that O B is p erp endi ular to O P .
p 1
6 Æ 6 Æ
Then sin e O B = 1, O P = 3, P B O = 60 and P ; B ; A are ollinear, B AO = 60
Æ1
implies that P O A = 30 . Therefore, each re e tion tak es the ra y around the ir le,6
so there are 11 re e tions.12