HMMT 十一月 2018 · 冲刺赛 · 第 8 题

HMMT November 2018 — Guts Round — Problem 8

[ 7 ] Pentagon JAM ES is such that AM = SJ and the internal angles satisfy ∠ J = ∠ A = ∠ E = 90 , and ∠ M = ∠ S . Given that there exists a diagonal of JAM ES that bisects its area, find the ratio of the shortest side of JAM ES to the longest side of JAM ES .

解析

[ 7 ] Pentagon JAM ES is such that AM = SJ and the internal angles satisfy \ J = \ A = \ E = 90 , and \ M = \ S . Given that there exists a diagonal of JAM ES that bisects its area, find the ratio of the shortest side of JAM ES to the longest side of JAM ES . Proposed by: James Lin 1 Answer: 4 Since \ J = \ A = 90 and AM = JS , JAM S must be a rectangle. In addition, \ M + \ S = 270 , so \ M = \ S = 135 . Therefore, \ ESM = \ EM S = 45 , which means M ES is an isosceles right triangle. Note that AM E and JSE are congruent, which means that [ JAES ] = [ JAE ] + [ JSE ] = [ JAE ] + [ AM E ] > [ AM E ], so AE cannot be our diagonal. Similarly, JE cannot be our diagonal. Diagonals SA and JM bisect rectangle JAM S , so they also cannot bisect the pentagon. Thus, the only diagonal that can bisect [ JAM ES ] is M S , which implies [ JAM S ] = [ M ES ]. We know M E · ES JA p [ JAM S ] = JA · AM and [ M ES ] = , and M E = ES = , which implies 2 2 2 JA AM 1 JA · AM = = ) = 4 JA 4 1 p Finally, EM and M S are both the length of SM = JA . This means that AM is our shortest side 2 1 and JA is our longest side, so is our answer. 4