HMMT 二月 2025 · 冲刺赛 · 第 4 题

HMMT February 2025 — Guts Round — Problem 4

[5] Let △ ABC be an equilateral triangle with side length 4. Across all points P inside triangle △ ABC satisfying [ P AB ] + [ P AC ] = [ P BC ], compute the minimum possible length of P A . (Here, [ XY Z ] denotes the area of triangle △ XY Z .) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2025, February 15, 2025 — GUTS ROUND Organization Team Team ID#

解析

[5] Let △ ABC be an equilateral triangle with side length 4. Across all points P inside triangle △ ABC satisfying [ P AB ] + [ P AC ] = [ P BC ], compute the minimum possible length of P A . (Here, [ XY Z ] denotes the area of triangle △ XY Z .) Proposed by: Isabella Zhu √ Answer: 3 Solution: A P B C The area condition implies [ ABC ] = 2[ P BC ]. Hence, P lies on the A -midline of △ ABC . Therefore, the minimum possible value of P A is the distance from A to this midline. This is achieved by taking P to be the foot of the perpendicular from A to the A -midline. This distance is half the altitude of √ √ 1 ABC , which has side length 4, so the answer is (2 3) = 3 . 2