HMMT 十一月 2011 · 冲刺赛 · 第 18 题

HMMT November 2011 — Guts Round — Problem 18

[ 10 ] Consider a cube ABCDEF GH , where ABCD and EF GH are faces, and segments AE, BF, CG, DH are edges of the cube. Let P be the center of face EF GH , and let O be the center of the cube. Given that AG = 1, determine the area of triangle AOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TH 4 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND Round 7

解析

[ 10 ] Consider a cube ABCDEF GH , where ABCD and EF GH are faces, and segments AE, BF, CG, DH are edges of the cube. Let P be the center of face EF GH , and let O be the center of the cube. Given that AG = 1, determine the area of triangle AOP . √ √ 2 1 2 √ √ Answer: From AG = 1, we get that AE = and AC = . We note that triangle AOP is 24 3 3 located in the plane of rectangle ACGE . Since OP ‖ CG and O is halway between AC and EG , we √ √ 1 1 1 2 2 √ √ get that [ AOP ] = [ ACGE ]. Hence, [ AOP ] = ( )( ) = . 8 8 24 3 3 A 1 √ 3 E B F √ 2 √ 3 O P D H C G