Školjka

%V0 A circle $C$ with center $O.$ and a line $L$ which does not touch circle $C.$ $OQ$ is perpendicular to $L,$ $Q$ is on $L.$ $P$ is on $L,$ draw two tangents $L_1, L_2$ to circle $C.$ $QA, QB$ are perpendicular to $L_1, L_2$ respectively. ($A$ on $L_1,$ $B$ on $L_2$). Prove that, line $AB$ intersect $QO$ at a fixed point. Original formulation: A line $l$ does not meet a circle $\omega$ with center $O.$ $E$ is the point on $l$ such that $OE$ is perpendicular to $l.$ $M$ is any point on $l$ other than $E.$ The tangents from $M$ to $\omega$ touch it at $A$ and $B.$ $C$ is the point on $MA$ such that $EC$ is perpendicular to $MA.$ $D$ is the point on $MB$ such that $ED$ is perpendicular to $MB.$ The line $CD$ cuts $OE$ at $F.$ Prove that the location of $F$ is independent of that of $M.$

%V0 Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

%V0 Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that $$\sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}.$$

%V0 Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

%V0 Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$. Let $AB$ be the diameter of $w$ perpendicular to $t$, so that $A, E, O_{1}$ are on the same side of $t$. Prove that lines $AO_{1}$, $BO_{2}$, $EF$ and $t$ are concurrent.

%V0 Consider five points $A$, $B$, $C$, $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$. Author: Charles Leytem, Luxembourg