HMMT 十一月 2014 · 冲刺赛 · 第 13 题

HMMT November 2014 — Guts Round — Problem 13

[ 9 ] Let ABC be a triangle with AB = AC = BC . Let M denote the midpoint of BC and let X 14 and Y denote the projections of M onto AB and AC , respectively. If the areas of triangle ABC and quadrilateral AXM Y are both positive integers, find the minimum possible sum of these areas.

解析

[ 9 ] Let ABC be a triangle with AB = AC = BC . Let M denote the midpoint of BC and let X 14 and Y denote the projections of M onto AB and AC , respectively. If the areas of triangle ABC and quadrilateral AXM Y are both positive integers, find the minimum possible sum of these areas. 24 2 Answer: 1201 By similar triangles, one can show that [ AXM Y ] = 2 · [ AM X ] = ( ) · 2[ ABM ] = 25 24 2 2 2 ( ) · [ ABC ]. Thus the answer is 25 + 24 = 1201. 25