Olympiad problems

Found problems: 2

1999 Korea Junior Math Olympiad, 1

There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.

2020 Brazil Team Selection Test, 3

Let $ABCD$ be a quadrilateral with a incircle $\omega$. Let $I$ be the center of $\omega$, suppose that the lines $AD$ and $BC$ intersect at $Q$ and the lines $AB$ and $CD$ intersect at $P$ with $B$ is in the segment $AP$ and $D$ is in the segment $AQ$. Let $X$ and $Y$ the incenters of $\triangle PBD$ and $\triangle QBD$ respectively. Let $R$ be the intersection of $PY$ and $QX$. Prove that the line $IR$ is perpendicular to $BD$.

geometric transformation incenter geometry inscribed quadrilateral projective geometry