Consider a rectangular table $2 \times n.$ Let every cell be dyed either by black or white color in a way that no $2\times 2$ square is completely black. Denote $P_n$ the number of such colorings. Prove that the number $P_{1989}$ is divisible by three and find the greatest power of three that divides them.

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.

The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.

What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?

The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

An equilateral triangle with side 101 is placed on a plane so that one of its sides is horizontal and the triangle is above it. It is divided into smaller equilateral triangles with side 1 by segments parallel to its sides. All sides of these smaller triangles are colored red (including the entire border of the large triangle). An equilateral triangle on a plane is called a "mirror" triangle if its sides are parallel to the sides of the original triangle, but it lies below its horizontal side. What is the smallest number of contours of mirror triangles needed to cover all the red segments? (Mirror triangles may overlap and extend beyond the original triangle.) [i]Proposed by Dmitry Shiryayev[/i]

$n$ points are marked on the board points that are vertices of the regular $n$ -gon. One of the points is a chip. Two players take turns moving it to the other marked point and at the same time draw a segment that connects them. If two points already connected by a segment, such a move is prohibited. A player who can't make a move, lose. Which of the players can guarantee victory?

Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.

On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?

Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Each vertex of a cube is assigned $1$ or $-1$. Each face is assigned the product of the four numbers at its vertices. Determine all possible values that can be obtained as the sum of all the $14$ assigned numbers.

One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.

A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.

a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have? b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for a) $N = 2011$ b) $N = 2012$ ?

There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?