Školjka

%V0 A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively). Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points. Original Statement: A circle $S$ is said to cut a circle $\Sigma$ diametrically if and only if their common chord is a diameter of $\Sigma.$ Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$

%V0 Let $n \geq 4$ be a fixed positive integer. Given a set $S = \{P_1, P_2, \ldots, P_n\}$ of $n$ points in the plane such that no three are collinear and no four concyclic, let $a_t,$ $1 \leq t \leq n,$ be the number of circles $P_iP_jP_k$ that contain $P_t$ in their interior, and let $m(S) = \sum^n_{i=1} a_i.$ Prove that there exists a positive integer $f(n),$ depending only on $n,$ such that the points of $S$ are the vertices of a convex polygon if and only if $m(S) = f(n).$

%V0 Given three fixed pairwisely distinct points $A$, $B$, $C$ lying on one straight line in this order. Let $G$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. The tangents to $G$ at $A$ and $C$ intersect each other at a point $P$. The segment $PB$ meets the circle $G$ at $Q$. Show that the point of intersection of the angle bisector of the angle $AQC$ with the line $AC$ does not depend on the choice of the circle $G$.

%V0 Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwisely distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.

%V0 Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

%V0 Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, assume that there exist points $E$ on line $BC$ outside segment $BC$, and $F$ inside segment $AD$ such that $\angle DAE = \angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$, assume it does not lie on line $AB$. Prove that $I$ belongs to the circumcircle of $ABK$ if and only if $K$ belongs to the circumcircle of $CDJ$. Proposed by Charles Leytem, Luxembourg