HMMT 十一月 2015 · 团队赛 · 第 1 题

英文解析

1. [ 3 ] Triangle ABC is isosceles, and ∠ ABC = x . If the sum of the possible measures of ∠ BAC is 240 ,
   find x .
   Proposed by: Alexander Katz
   Answer: 20
   There are three possible triangles: either ∠ ABC = ∠ BCA , in which case ∠ BAC = 180 − 2 x , ∠ ABC =
   180 − x
   ∠ BAC , in which case ∠ BAC = x , or ∠ BAC = ∠ BCA , in which case ∠ BAC = . These sum to
   540 − 3 x 540 − 3 x 2
   , so we have = 240 = ⇒ x = 20 .
   2 2