PUMaC 2016 · 几何（A 组） · 第 6 题
PUMaC 2016 — Geometry (Division A) — Problem 6
题目详情
- In isosceles triangle ABC with base BC , let M be the midpoint of BC . Let P be the inter- section of the circumcircle of 4 ACM with the circle with center B passing through M , such √ CP ◦ that P 6 = M . If ∠ BP C = 135 , then can be written as a + b for positive integers a and AP b , where b is not divisible by the square of any prime. Find a + b .
解析
- Let BP = 1 , CP = y, AP = x . By the law of cosines we have: √ 2 2 2 2 2 1 + x + 2 · 1 · x = BC = (2 · BP ) = 2 √ 2 2 2 2 2 2 1 + y + 2 · 1 · y = AB = AC = x + y √ √ √ 7 − 1 3 − 7 CP x √ √ Solving (using the fact x > 0) yields x = and y = . Then, = = 2 + 7, so AP y 2 2 a + b = 9 . Problem written by Bill Huang.