What is the smallest number of squares of a chess board that can be marked in such a manner that (a) no two marked squares may have a common side or a common vertex, and (b) any unmarked square has a common side or a common vertex with at least one marked square? Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice. (A. Andjans, Riga)

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.

It is intended to make a map locating $30$ different cities on it. For this, all the distances between these cities are available as data (each of these distances is considered as a “data”). Three of these cities are already laid out on the map, and they turn out to be non-collinear. How much data must be used as a minimum to complete the map?

Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points. (Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)

Given $n$ points in the three dimensional space such, that the arbitrary triangle with the vertices in three of those points contains an angle greater than $120$ degrees. Prove that you can rearrange them to make a polyline (unclosed) with all the angles between the sequent links greater than $120$ degrees.

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$.

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply: (a) the circumcircle of each triangle in the set $T$ is $\omega$; (b) The interior of any two triangles in the set $T$ has no common point. Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.

The rectangular table has four rows. The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number $n$ and write the number $k$ if $n$ was met $k$ times. Prove that the second row coincides with the fourth one.

We are given $2001$ lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called [i]sides[/i], and the collection of the regions is called a [i]map[/i]. Intersection points of the lines are called [i]vertices[/i]. Two regions are [i]neighbors [/i]if they share a side, and two vertices are neighbors if they lie on the same side. A [i]legal coloring[/i] of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?

Determine all integers $m$, for which it is possible to dissect the square $m\times m$ into five rectangles, with the side lengths being the integers $1, 2, … ,10$ in some order.

Given a finite set $B$ of points in space, any two distances between the points of this set are different. Each point of the set $B$ is connected by a line segment to the closest point of the set $B$. This way we will get a set of sections, one of which (any chosen one) we paint red, all the remaining sections we paint green. Prove that there are two points of the set $B$ that cannot be connected by a line composed of green segments.

On the ends of the diameter two "$1$"s are written. Each of the semicircles is divided onto two parts and the sum of the numbers of its ends (i.e. "$2$") is written at the midpoint. Then every of the four arcs is halved and in its midpoint the sum of the numbers on its ends is written. Find the total sum of the numbers on the circumference after $n$ steps.

Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.

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.

Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside. (AV Shapovelov)

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

Let $S$ be a set of $2020$ distinct points in the plane. Let \[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\] Find the least possible value of the number of points in $M$.