PUMaC 2013 · 团队赛 · 第 11 题

PUMaC 2013 — Team Round — Problem 11

If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?

解析

If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle? SOLUTION: Suppose the centre of the circle is O and one endpoint A of the chord is fixed. ◦ ◦ The other endpoint B of the chord must satisfy ∠ AOB > 60 . This means there is a 120 240 2 ′ ′′ sector ∠ B OB where the other point cannot be. Hence the answer is = . 360 3 2 ANSWER: . 3