HMMT 二月 2007 · 冲刺赛 · 第 16 题
HMMT February 2007 — Guts Round — Problem 16
题目详情
- [ 10 ] Let ABC be a triangle with AB = 7 , BC = 9 , and CA = 4. Let D be the point such that
AB ‖ CD and CA ‖ BD . Let R be a point within triangle BCD . Lines
and m going through R are ′ parallel to CA and AB respectively. Linemeets AB and BC at P and P respectively, and m meets ′ CA and BC at Q and Q respectively. If S denotes the largest possible sum of the areas of triangles ′ ′ ′ ′ 2 BP P , RP Q , and CQQ , determine the value of S .
解析
- [ 10 ] Let ABC be a triangle with AB = 7 , BC = 9 , and CA = 4. Let D be the point such that
AB ‖ CD and CA ‖ BD . Let R be a point within triangle BCD . Lines
and m going through R are ′ parallel to CA and AB respectively. Linemeets AB and BC at P and P respectively, and m meets ′ CA and BC at Q and Q respectively. If S denotes the largest possible sum of the areas of triangles ′ ′ ′ ′ 2 BP P , RP Q , and CQQ , determine the value of S . 4 ′ ′ ′ Answer: 180 . Let R denote the intersection of the lines through Q and P parallel to ` and m ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ respectively. Then [ RP Q ] = [ R P Q ]. Triangles BP P , R P Q , and CQQ lie in ABC without overlap, so that on the one hand, S ≤ ABC . On the other, this bound is realizable by taking R to be a vertex of triangle BCD . We compute the square of the area of ABC to be 10 · (10 − 9) · (10 − 7) · (10 − 4) =