HMMT 十一月 2023 · THM 赛 · 第 9 题

HMMT November 2023 — THM Round — Problem 9

Pentagon SP EAK is inscribed in triangle N OW such that S and P lie on segment N O , K and A lie on segment N W , and E lies on segment OW . Suppose that N S = SP = P O and N K = KA = AW . Given that EP = EK = 5 and EA = ES = 6 , compute OW .

解析

Pentagon SP EAK is inscribed in triangle N OW such that S and P lie on segment N O , K and A lie on segment N W , and E lies on segment OW . Suppose that N S = SP = P O and N K = KA = AW . Given that EP = EK = 5 and EA = ES = 6, compute OW . Proposed by: Rishabh Das q √ 3 610 122 Answer: = 3 5 5 Solution: N K S 6 A P 5 5 6 W O E Note that [ ESK ] = [ EP A ], since one has half the base but double the height. Since the sides are the ◦ same, we must have sin ∠ SEK = sin ∠ P EA , so ∠ SEK + ∠ P EA = 180 . Let OW = 3 x , so SK = x and P A = 2 x . Then by the law of cosines 2 x = 61 − 60 cos ∠ SEK 2 4 x = 61 − 60 cos ∠ P EA. q 122 2 Summing these two gives 5 x = 122, since cos ∠ SEK = − cos ∠ P EA . Then x = , which means 5 √ 3 610 3 x = . 5