HMMT 十一月 2015 · 团队赛 · 第 7 题
HMMT November 2015 — Team Round — Problem 7
题目详情
英文原题
- [ 6 ] Let ABCD be a convex quadrilateral whose diagonals AC and BD meet at P . Let the area oftriangle AP B be 24 and let the area of triangle CP D be 25. What is the minimum possible area ofquadrilateral ABCD ?
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解析
英文解析
- [ 6 ] Let ABCD be a convex quadrilateral whose diagonals AC and BD meet at P . Let the area oftriangle AP B be 24 and let the area of triangle CP D be 25. What is the minimum possible area ofquadrilateral ABCD ?
Proposed by: Sam Korsky
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Answer: 49 + 20 6
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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
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[ BP C ] + [ DP A ] ≥ 2 [ BP C ][ DP A ] = 20 6
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so the minimum area of quadrilateral ABCD is 49 + 20 6 .
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