HMMT 十一月 2015 · 团队赛 · 第 7 题

HMMT November 2015 — Team Round — Problem 7

[ 6 ] Let ABCD be a convex quadrilateral whose diagonals AC and BD meet at P . Let the area of triangle AP B be 24 and let the area of triangle CP D be 25. What is the minimum possible area of quadrilateral ABCD ? 3 4 5 11 5

解析

[ 6 ] Let ABCD be a convex quadrilateral whose diagonals AC and BD meet at P . Let the area of triangle AP B be 24 and let the area of triangle CP D be 25. What is the minimum possible area of quadrilateral ABCD ? Proposed by: Sam Korsky √ Answer: 49 + 20 6 ◦ ◦ Note that ∠ AP B = 180 − ∠ BP C = ∠ CP D = 180 − ∠ DP A so 4[ BP C ][ DP A ] = ( P B · P C · sin BP C )( P D · P A · sin DP A ) = ( P A · P B · sin AP B )( P C · P D · sin CP D ) = 4[ AP B ][ CP D ] = 2400 = ⇒ [ BP C ][ DP A ] = 600. Hence by AM-GM we have that √ √ [ BP C ] + [ DP A ] ≥ 2 [ BP C ][ DP A ] = 20 6 √ so the minimum area of quadrilateral ABCD is 49 + 20 6 . 3 4 5 11 5