Školjka

%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 A number of $n$ rectangles are drawn in the plane. Each rectangle has parallel sides and the sides of distinct rectangles lie on distinct lines. The rectangles divide the plane into a number of regions. For each region $R$ let $v(R)$ be the number of vertices. Take the sum $\sum v(R)$ over the regions which have one or more vertices of the rectangles in their boundary. Show that this sum is less than $40n$.

%V0 Let $A_1A_2 \ldots A_n$ be a convex polygon, $n \geq 4.$ Prove that $A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $A_j$ one can assign a pair $(b_j, c_j)$ of real numbers, $j = 1, 2, \ldots, n,$ so that $A_iA_j = b_jc_i - b_ic_j$ for all $i, j$ with $1 \leq i < j \leq n.$

%V0 Let $ABC$ be an acute-angled triangle, and let $w$ be the circumcircle of triangle $ABC$. The tangent to the circle $w$ at the point $A$ meets the tangent to the circle $w$ at $C$ at the point $B^{\prime}$. The line $BB^{\prime}$ intersects the line $AC$ at $E$, and $N$ is the midpoint of the segment $BE$. Similarly, the tangent to the circle $w$ at the point $B$ meets the tangent to the circle $w$ at the point $C$ at the point $A^{\prime}$. The line $AA^{\prime}$ intersects the line $BC$ at $D$, and $M$ is the midpoint of the segment $AD$. a) Show that $\measuredangle ABM = \measuredangle BAN$. b) If $AB = 1$, determine the values of $BC$ and $AC$ for the triangles $ABC$ which maximise $\measuredangle ABM$.

%V0 Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?

%V0 Prove that there exist infinitely many positive integers $n$ such that $p = nr,$ where $p$ and $r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.