$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

The shape of a military base is an equilateral triangle of side $10$ kilometers. Security constraints make cellular phone com­munication possible only within $2.5$ kilometers. Each of $17$ soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.

There are given 11 distinct points inside a ball with volume $V.$ Show that there are two planes $\varrho,\sigma,$ both containing the center of the ball, such that the resulting spherical wedge has volume $V/8$ and its interior contains none of the given points.

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.

Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?

We have an $8\times 8$ board. An [i]interior [/i] edge is an edge between two $1 \times 1$ cells. we cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board so that it cuts along edge $k$. Calculate the last digit of the sum we get if we add all $N(k)$, where $k$ is an inner edge.

On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way. [i]Attention: you have to really prove that a greater number of dominoes is impossible. [/i]

The vertices of a regular $2002$-sided polygon are numbered $1$ through $2002$, clockwise. Given an integer $ n$, $1 \le n \le 2002$, color vertex $n$ blue, then, going clockwise, count$ n$ vertices starting at the next of $n$, and color $n$ blue. And so on, starting from the vertex that follows the last vertex that was colored, n vertices are counted, colored or uncolored, and the number $n$ is colored blue. When the vertex to be colored is already blue, the process stops. We denote $P(n)$ to the set of blue vertices obtained with this procedure when starting with vertex $n$. For example, $P(364)$ is made up of vertices $364$, $728$, $1092$, $1456$, $1820$, $182$, $546$, $910$, $1274$, $1638$, and $2002$. Determine all integers $n$, $1 \le n \le 2002$, such that $P(n)$ has exactly $14 $ vertices,

a) A bus network is organized so that: 1) one can reach any stop from any other stop without changing buses; 2) every pair of routes has a single stop at which one can change buses; 3) each route has exactly three stops? How many bus routes are there? It is assumed that there are at least two routes. b) A town has $57$ bus routes. How many stops does each route have if it is known that 1) one can reach any stop from any other stop without changing buses; 2) for every pair of routes there is a single stop where one can change buses; 3) each route has three or more stops?

A square was split into several rectangles so that the centers of rectangles form a convex polygon. a) Is it true for sure that each rectangle adjoins to a side of the square? b) Can the number of rectangles equal 23 ? Alexandr Shapovalov

A convex $n$-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that $n$ must be a multiple of $3$.

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area $3$, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.

Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

Sixteen squares of an $8\times 8$ chessboard are chosen so that there are exactly lwo in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in cach colunm.

Let $n$ be a positive integer. Given $n$ points in the plane, prove that it is possible to draw an angle with measure $\frac{2\pi}{n}$ with vertex as each one of the given points, such that any point in the plane is covered by at least one of the angles.

A configuration of several checkers at the centers of squares on a rectangular sheet of grid paper is called [i]boring [/i] if some four checkers occupy the vertices of a rectangle with sides parallel to those of the sheet. (a) Prove that any configuration of more than $3mn/4$ checkers on an $m\times n$ grid is boring. (b) Prove that any configuration of $26$ checkers on a $7\times 7$ grid is boring.

Fix an integer $n \geq 3$. Let $\mathcal{S}$ be a set of $n$ points in the plane, no three of which are collinear. Given different points $A,B,C$ in $\mathcal{S}$, the triangle $ABC$ is [i]nice[/i] for $AB$ if $[ABC] \leq [ABX]$ for all $X$ in $\mathcal{S}$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several nice triangles). A triangle is [i] beautiful [/i] if its vertices are all in $\mathcal{S}$ and is nice for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ beautiful triangles.

There are nine points in the data square, of which no three are collinear. Prove that three of them are vertices of a triangle with an area not exceeding $ \frac{1}{8} $ the area of a square.

$D$ is a closed disk radius $R$. Show that among any $8$ points of $D$ one can always find two whose distance apart is less than $R$.

Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.