Olympiad problems

2007 Romania Team Selection Test, 2

Let $ABC$ be a triangle, and $\omega_{a}$, $\omega_{b}$, $\omega_{c}$ be circles inside $ABC$, that are tangent (externally) one to each other, such that $\omega_{a}$ is tangent to $AB$ and $AC$, $\omega_{b}$ is tangent to $BA$ and $BC$, and $\omega_{c}$ is tangent to $CA$ and $CB$. Let $D$ be the common point of $\omega_{b}$ and $\omega_{c}$, $E$ the common point of $\omega_{c}$ and $\omega_{a}$, and $F$ the common point of $\omega_{a}$ and $\omega_{b}$. Show that the lines $AD$, $BE$ and $CF$ have a common point.

2020 Iran RMM TST, 4

Tags: geometry , moving points , desargue

In a trapezoid $ABCD$ with $AD$ parallel to $BC$ points $E, F$ are on sides $AB, CD$ respectively. $A_1, C_1$ are on $AD,BC$ such that $A_1, E, F, A$ lie on a circle and so do $C_1, E, F, C$. Prove that lines $A_1C_1, BD, EF$ are concurrent.

2004 USA Team Selection Test, 4

Tags: circumcircle , geometry , desargue , parallel lines , power of a point

Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.

2013 Saudi Arabia IMO TST, 1

Tags: concurrency , incircle , desargue , pascal , tangent , geometry , collinear

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.