HMMT 十一月 2014 · 冲刺赛 · 第 13 题
HMMT November 2014 — Guts Round — Problem 13
题目详情
英文原题
- [ 9 ] Let ABC be a triangle with AB = AC = BC . Let M denote the midpoint of BC and let Xand Y denote the projections of M onto AB and AC , respectively. If the areas of triangle ABC and 14
quadrilateral AXM Y are both positive integers, find the minimum possible sum of these areas.
解析
英文解析
- [ 9 ] Let ABC be a triangle with AB = AC = BC . Let M denote the midpoint of BC and let Xand Y denote the projections of M onto AB and AC , respectively. If the areas of triangle ABC and 14
quadrilateral AXM Y are both positive integers, find the minimum possible sum of these areas.
Answer: 1201 By similar triangles, one can show that [ AXM Y ] = 2 · [ AM X ] = ( ) · 2[ ABM ] =224
2425
2 2 2
( ) · [ ABC ]. Thus the answer is 25 + 24 = 1201.
25