HMMT 十一月 2021 · GEN 赛 · 第 3 题

HMMT November 2021 — GEN Round — Problem 3

Let ABCD be a unit square. A circle with radius passes through point D and is tangent to side 49 m AB at point E . Then DE = , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n

解析

Let ABCD be a unit square. A circle with radius passes through point D and is tangent to side 49 m AB at point E . Then DE = , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n Proposed by: David Vulakh Answer: 807 Solution: F D C O A B E Let O be the center of the circle and let F be the intersection of lines OE and CD . Also let r = 32 / 49 and x = DF . Then we know 2 2 2 2 2 2 x + (1 − r ) = DF + OF = DO = r , 2 2 which implies that x + 1 − 2 r = 0, or 1 + x = 2 r . Now, √ √ √ √ 2 2 2 DE = DF + EF = 1 + x = 2 r = 64 / 49 = 8 / 7 .