HMMT 二月 2021 · 冲刺赛 · 第 18 题

HMMT February 2021 — Guts Round — Problem 18

[12] Triangle ABC has side lengths AB = 19, BC = 20, and CA = 21. Points X and Y are selected on sides AB and AC , respectively, such that AY = XY and XY is tangent to the incircle of 4 ABC . If the a length of segment AX can be written as , where a and b are relatively prime positive integers, compute b 100 a + b .

解析

[12] Triangle ABC has side lengths AB = 19, BC = 20, and CA = 21. Points X and Y are selected on sides AB and AC , respectively, such that AY = XY and XY is tangent to the incircle of 4 ABC . a If the length of segment AX can be written as , where a and b are relatively prime positive integers, b compute 100 a + b . Proposed by: Hahn Lheem Answer: 6710 Solution: Note that the incircle of 4 ABC is the A -excenter of 4 AXY . Let r be the radius of this circle. We can compute the area of 4 AXY in two ways: 1 K = · AX · AY sin A AXY 2 = r · ( AX + AY − XY ) / 2 r = ⇒ AY = sin A We also know that 1 K = · 19 · 21 sin A ABC 2 = r · (19 + 20 + 21) / 2 r 19 · 21 133 = ⇒ = = sin A 60 20 so AY = 133 / 20. Let the incircle of 4 ABC be tangent to AB and AC at D and E , respectively. We know that 133 67 AX + AY + XY = AD + AE = 19 + 21 − 20, so AX = 20 − = . 10 10