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Rude quadrilateral

Source: Serbian Mathematical Olympiad 2021, P4

May 14, 2021
geometrySerbia

Problem Statement

A convex quadrilateral ABCDABCD will be called rude if there exists a convex quadrilateral PQRSPQRS whose points are all in the interior or on the sides of quadrilateral ABCDABCD such that the sum of diagonals of PQRSPQRS is larger than the sum of diagonals of ABCDABCD.
Let r>0r>0 be a real number. Let us assume that a convex quadrilateral ABCDABCD is not rude, but every quadrilateral A′BCDA'BCD such that A′≠AA'\neq A and A′A≤rA'A\leq r is rude. Find all possible values of the largest angle of ABCDABCD.
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