Olympiad problems

2015 Middle European Mathematical Olympiad, 3

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.

2021 Peru Iberoamerican Team Selection Test, P4

Tags: ISL 2021 , IMO Shortlist , geometry , cyclic quadrilateral , concurrency , Angle Chasing , imo shortlist g4

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2023 Rioplatense Mathematical Olympiad, 2

Tags: geometry , excenter , Angle Chasing

Let $ABCD$ be a convex quadrilateral, such that $AB = CD$, $\angle BCD = 2 \angle BAD$, $\angle ABC = 2 \angle ADC$ and $\angle BAD \neq \angle ADC$. Determine the measure of the angle between the diagonals $AC$ and $BD$.

2022 Brazil National Olympiad, 2

Tags: geometry , Angle Chasing , geometric transformation , reflection

Let $ABC$ be an acute triangle, with $AB<AC$. Let $K$ be the midpoint of the arch $BC$ that does not contain $A$ and let $P$ be the midpoint of $BC$. Let $I_B,I_C$ be the $B$-excenter and $C$-excenter of $ABC$, respectively. Let $Q$ be the reflection of $K$ with respect to $A$. Prove that the points $P,Q,I_B,I_C$ are concyclic.

2018 India PRMO, 29

Tags: geometry , incenter , Angle Chasing

Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively. If $\angle BI_1E = 60^o$, what is the measure of $\angle CI_2F$ in degrees?