PUMaC 2017 · 团队赛 · 第 6 题

PUMaC 2017 — Team Round — Problem 6

(5) In regular pentagon ABCDE , let O ∈ CE be the center of circle Γ tangent to DA and DE . Γ meets DE at X and DA at Y . Let the altitude from B meet CD at P ; if CP = 1, the ◦ a sin c area of 4 COY can be written in the form , where a and b are relatively prime positive 2 ◦ b cos c integers and c is an integer in the range (0 , 90). Find a + b + c .

解析

First, assume the pentagon has side length 1; at the end we will divide our value for 4 COY ’s 2 area by CP (as area is quadratic in length). We note that 4 COD is a 36 − 54 − 90 triangle, so 1 CO = and DO = tan 36. 4 DOX is an 18 − 72 − 90 triangle so OX = OY = sin 18 tan 36. cos 36 2 1 1 sin 18 sin 36 Thus the desired area is, since ∠ COY = 18, CO · OY sin 18 = . ∠ P CB = 72 so 2 2 2 cos 36 1 sin 36 ∠ P BC = 18 and thus P B = sin 18; the area now in the form so the final answer is 2 2 cos 36 1 + 2 + 36 = 39 . Problem written by Zack Stier 2017