Olympiad problems

AND:

OR:

NO:

Found problems: 2

2025 Israel TST, P2

Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.

2023 Serbia National Math Olympiad, 6

Tags: geometry , mixtilinear incircle , configurations

Given is a triangle $ABC$ with incenter $I$ and circumcircle $\omega$. The incircle is tangent to $BC$ at $D$. The perpendicular at $I$ to $AI$ meets $AB, AC$ at $E, F$ and the circle $(AEF)$ meets $\omega$ and $AI$ at $G, H$. The tangent at $G$ to $\omega$ meets $BC$ at $J$ and $AJ$ meets $\omega$ at $K$. Prove that $(DJK)$ and $(GIH)$ are tangent to each other.