Školjka

%V0 What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? Posted already on the board I think...

%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 $S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $P$ whose vertices are in $S$, let $a(P)$ be the number of vertices of $P$, and let $b(P)$ be the number of points of $S$ which are outside $P$. A line segment, a point, and the empty set are considered as convex polygons of $2$, $1$, and $0$ vertices respectively. Prove that for every real number $x$: $\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1$, where the sum is taken over all convex polygons with vertices in $S$. Alternative formulation: Let $M$ be a finite point set in the plane and no three points are collinear. A subset $A$ of $M$ will be called round if its elements is the set of vertices of a convex $A -$gon $V(A).$ For each round subset let $r(A)$ be the number of points from $M$ which are exterior from the convex $A -$gon $V(A).$ Subsets with $0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $A$ of $M$ construct the polynomial $$P_A(x) = x^{|A|}(1 - x)^{r(A)}.$$ Show that the sum of polynomials for all round subsets is exactly the polynomial $P(x) = 1.$

%V0 A cake has the form of an $n$ x $n$ square composed of $n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $A$. Let $B$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $B$ than of arrangement $A$. Prove that arrangement $B$ can be obtained from $A$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

%V0 Consider a convex pentagon $ABCDE$ such that $$\angle BAC = \angle CAD = \angle DAE\ \ \ ,\ \ \ \angle ABC = \angle ACD = \angle ADE$$ Let $P$ be the point of intersection of the lines $BD$ and $CE$. Prove that the line $AP$ passes through the midpoint of the side $CD$.

%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