PUMaC 2017 · 几何（B 组） · 第 2 题
PUMaC 2017 — Geometry (Division B) — Problem 2
题目详情
- A kite is inscribed in a circle with center O and radius 60. The diagonals of the kite meet at a point P , and OP is an integer. The minimum possible area of the kite can be expressed in √ the form a b , where a and b are positive integers and b is squarefree. Find a + b .
解析
- Let OP = k . Then (60 + k )(60 − k ) = , where d is the length of the other diagonal of the 2 2 kite. But (60 + k )(60 − k ) = 3600 − k is minimized and positive when k = 59 for k an integer. √ √ 1 Then the area of the kite is · 120 · 2 119 = 120 119, so the answer is 239 . 2 Problem written by Nathan Bergman ◦