Olympiad problems

2024 USEMO, 3

Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that \[ \angle API = \angle AQI = 45^\circ. \] Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$. [i]Matsvei Zorka[/i]

2018 Taiwan TST Round 1, 6

Given six points $ A, B, C, D, E, F $ such that $ \triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA $ and let $ I $ be the incenter of $ \triangle ABC. $ Prove that the circumcenter of $ \triangle AID, \triangle BIE, \triangle CIF $ are collinear. [i]Proposed by Telv Cohl[/i]

incenter geometry inversion coaxial circles geometric inversion radical axis spiral similarity

2021 Brazil National Olympiad, 6

Let $n \geq 5$ be integer. The convex polygon $P = A_{1} A_{2} \ldots A_{n}$ is bicentric, that is, it has an inscribed and circumscribed circle. Set $A_{i+n}=A_{i}$ to every integer $i$ (that is, all indices are taken modulo $n$). Suppose that for all $i, 1 \leq i \leq n$, the rays $A_{i-1} A_{i}$ and $A_{i+2} A_{i+1}$ meet at the point $B_{i}$. Let $\omega_{i}$ be the circumcircle of $B_{i} A_{i} A_{i+1}$. Prove that there is a circle tangent to all $n$ circles $\omega_{i}$, $1 \leq i \leq n$.

convex polygon inscribed circle tangent circle geometry circumscribed bicentric polygon angle chasing geometric inversion homothety radical axis

Mexican Quarantine Mathematical Olympiad, #3

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. [i]Proposed by Ariel García[/i]

concyclic tangent geometry angle chasing cyclic quadrilaterals geometric inversion homothety law of sines power of a point radical axis symmedians