$N$ lines are drawn on the plane that divide it into a certain number for finite and endless parts. For which number of straight lines $n$ can there be more finite than infinite among the resulting level parts?

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

Inside a convex $1000$-gon, $500$ points are selected so that no three of the $1500$ points — the ones selected and the vertices of the polygon — lie on the same straight line. This $1000$-gon is then divided into triangles so that all $1500$ points are vertices of the triangles, and so that these triangles have no other vertices. How many triangles will there be?

Every point of the plane is painted with one of three colors. Can we always find two points a distance $1$ cm apart which are of the same color?

$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.

Suppose we have $2016$ points in a $2$-dimensional plane such that no three lie on a line. Two quadrilaterals are not disjoint if they share an edge or vertex, or if their edges intersect. Show that there are at least $504$ quadrilaterals with vertices among these points such that any two of the quadrilaterals are disjoint.

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

Prove that in each polyhedron there exist two faces with the same number of edges.

A cube of dimensions $20 \times 20 \times 20$ is constructed with blocks of $1 \times 2 \times 2$. Prove that there is a line that passes through the cube but not any block.

There are $2023$ guinea pigs placed in a circle, from which everyone except one of them, call it $M$, has a mirror that points towards one of the $2022$ other guinea pigs. $M$ has a lantern that will shoot a light beam towards one of the guinea pigs with a mirror and will reflect to the guinea pig that the mirror is pointing and will keep reflecting with every mirror it reaches. Isaías will re-direct some of the mirrors to point to some other of the $2023$ guinea pigs. In the worst case scenario, what is the least number of mirrors that need to be re-directed, such that the light beam hits $M$ no matter the starting point of the light beam?

$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.

There are $n^2$ segments in the plane (read walls), no two of which are parallel or intersecting. Prove that there are at least $n$ points in the plane such that no two of them see each other (meaning there is a wall separating them).

In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$. In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.

A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called [i]good [/i] if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is [i]good[/i].

The vertices of a convex $2550$-gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white. [i]D. Rudenko[/i]

An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?

Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.

It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that: $i)$ sum of lengths of all diameters of all circles is not greater than $2018$ $ii)$ distance between any two circles is greater than $1$

To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?