HMMT 二月 2000 · 几何 · 第 10 题
HMMT February 2000 — Geometry — Problem 10
题目详情
- Let C and C b e t w o on en tri re e tiv e hollo w metal spheres of radius R and R 3 1 2 Æ 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 30 2 from the radial dire tion. The ra y ev en tually returns to P . Ho w man y total re e tions o of C and C do es it tak e? 1 2
解析
- Consider the attened v ersion of the situation. Then let O b e the en ter fo the spheres, A b e the rst re e tion p oin t, B b e the p oin t of C su h that O B is p erp endi ular to O P . 1 p 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 1 Æ 6 implies that P O A = 30 . Therefore, ea h re e tion tak es the ra y around the ir le, 12 so there are 11 re e tions.