HMMT 十一月 2023 · GEN 赛 · 第 6 题
HMMT November 2023 — GEN Round — Problem 6
题目详情
- Let ABCD be a square of side length 5 . A circle passing through A is tangent to segment CD at T and meets AB and AD again at X ̸ = A and Y ̸ = A , respectively. Given that XY = 6 , compute AT .
解析
- Let ABCD be a square of side length 5 . A circle passing through A is tangent to segment CD at T and meets AB and AD again at X ̸ = A and Y ̸ = A , respectively. Given that XY = 6 , compute AT . Proposed by: Maxim Li √ Answer: 30 Solution: X A B O Z Y D C T Let O be the center of the circle, and let Z be the foot from O to AD . Since XY is a diameter, √ √ √ 2 OT = ZD = 3 , so AZ = 2 . Then OZ = 5 and AT = OZ + 25 = 30 .