HMMT 十一月 2019 · 冲刺赛 · 第 13 题

HMMT November 2019 — Guts Round — Problem 13

[9] In 4 ABC , the incircle centered at I touches sides AB and BC at X and Y , respectively. Additionally, 2 the area of quadrilateral BXIY is of the area of ABC . Let p be the smallest possible perimeter of a 5 4 ABC that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter p .

解析

[9] In 4 ABC , the incircle centered at I touches sides AB and BC at X and Y , respectively. Addition- 2 ally, the area of quadrilateral BXIY is of the area of ABC . Let p be the smallest possible perimeter 5 of a 4 ABC that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter p . Proposed by: Joey Heerens √ Answer: 2 5 Note that ∠ BXI = ∠ BY I = 90, which means that AB and BC are tangent to the incircle of ABC [ BXIY ] AB + BC − AC 2 AB + BC − AC at X and Y respectively. So BX = BY = , which means that = = . 2 5 [ ABC ] AB + BC + AC The smallest perimeter is achieved when AB = AC = 3 and BC = 4. The area of this triangle ABC √ is 2 5.