Olympiad problems

2019 IMO Shortlist, G7

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$. [i]Proposed by Anant Mudgal, India[/i]

2007 IMO Shortlist, 2

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

cyclic quadrilateral circumcircle geometry mixtilinear incircle reflection angle chasing cyclic quadrilaterals symmedians

2019 Peru IMO TST, 3

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

incenter circumcircle geometry mixtilinear incircle