Found problems: 478
Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$.
Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively.
Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively.
Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.
Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent.
(S. Arkhipov)
Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$ in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent.
(D.Shvetsov, A.Zaslavsky)