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Source: China TST 2023 Problem 15

March 27, 2023
geometry

Problem Statement

For a convex quadrilateral ABCDABCD, call a point in the interior of ABCDABCD balanced, if
(1) PP is not on AC,BDAC,BD
(2) Let AP,BP,CP,DPAP,BP,CP,DP intersect the boundaries of ABCDABCD at A′,B′,C′,D′A', B', C', D', respectively, then
AP⋅PA′=BP⋅PB′=CP⋅PC′=DP⋅PD′AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'
Find the maximum possible number of balanced points.
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