HMMT 二月 2002 · 冲刺赛 · 第 39 题
HMMT February 2002 — Guts Round — Problem 39
题目详情
英文原题
- [7] In the x - y plane, draw a circle of radius 2 centered at (0 , 0). Color the circle redabove the line y = 1, color the circle blue below the line y = − 1, and color the rest of the circle white. Now consider an arbitrary straight line at distance 1 from the circle. We coloreach point P of the line with the color of the closest point to P on the circle. If we pick suchan arbitrary line, randomly oriented, what is the probability that it contains red, white, andblue points?
√
2 2
解析
英文解析
- In the x - y plane, draw a circle of radius 2 centered at (0 , 0). Color the circle redabove the line y = 1, color the circle blue below the line y = − 1, and color the rest of the circle white. Now consider an arbitrary straight line at distance 1 from the circle. We coloreach point P of the line with the color of the closest point to P on the circle. If we pick suchan arbitrary line, randomly oriented, what is the probability that it contains red, white, andblue points?
Solution: Let O = (0 , 0) , P = (1 , 0), and H the foot of the perpendicular from O tothe line. If ∠ P OH (as measured counterclockwise) lies between π/ 3 and 2 π/ 3, the line willfail to contain blue points; if it lies between 4 π/ 3 and 5 π/ 3, the line will fail to contain red
2 π 2
points. Otherwise, it has points of every color. Thus, the answer is 1 − / 2 π = .
3 3
√
2 2