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HMMT 二月 2008 · TEAM2 赛 · 第 10 题

HMMT February 2008 — TEAM2 Round — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Let a, b, c denote the side lengths of BC, CA, AB . Find the lengths of AE, BF, CD in terms of a, b, c .
解析
  1. [ 15 ] Let a, b, c denote the side lengths of BC, CA, AB . Find the lengths of AE, BF, CD in terms of a, b, c . Solution: Let x = AE = AF , y = BD = BF , z = CD = CE . Since BC = BD + CD , we have a = x + y . Similarly with the other sides, we arrive at the following system of equations: a = y + z, b = x + z, c = x + y. Solving this system gives us b + c − a AE = x = , 2 a + c − b BF = y = , 2 a + b − c CD = z = . 2