HMMT 十一月 2018 · THM 赛 · 第 7 题

HMMT November 2018 — THM Round — Problem 7

Ben “One Hunna Dolla” Franklin is flying a kite KIT E such that IE is the perpendicular bisector of KT . Let IE meet KT at R . The midpoints of KI, IT, T E, EK are A, N, M, D , respectively. Given that [ M AKE ] = 18 , IT = 10 , [ RAIN ] = 4, find [ DIM E ]. Note: [ X ] denotes the area of the figure X .

解析

Ben “One Hunna Dolla” Franklin is flying a kite KIT E such that IE is the perpendicular bisector of KT . Let IE meet KT at R . The midpoints of KI, IT, T E, EK are A, N, M, D , respectively. Given that [ M AKE ] = 18 , IT = 10 , [ RAIN ] = 4, find [ DIM E ]. Note: [ X ] denotes the area of the figure X . Proposed by: Michael Ren Answer: 16 Let [ KIR ] = [ RIT ] = a and [ KER ] = [ T ER ] = b . We will relate all areas to a and b . First, 1 1 [ RAIN ] = [ RAI ] + [ IN R ] = a + a = a. 2 2 Next, we break up [ M AKE ] = [ M AD ] + [ AKD ] + [ DEM ]. We have AD · DM 1 IE KT [ KIT E ] a + b [ M AD ] = = · · = = 2 2 2 2 4 2 [ KIE ] a + b [ AKD ] = = 4 4 [ KT E ] b [ DEM ] = = . 4 2 3 a +5 b After adding these we get [ M AKE ] = . We want to find 4 ( ) 4 3 a + 5 b 2 4 2 [ DIM E ] = 2[ IM E ] = [ IT E ] = a + b = + a = · 18 + · 4 = 16 . 5 4 5 5 5