Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.

Consider a convex $15$- gon with perimeter $21$. Show that there one can select three distinct pairs of vertices that form a triangle with area less than $1$. [hide=original wording of second sentence]Zeige, dass man davon drei paarweise verschiedene Eckpunkte auswählen kann, die ein Dreieck mit Fläche kleiner als 1 bilden.[/hide]

Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

A cube has been divided into $27$ equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?

Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?

We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.

A table is covered by $15$ pieces of paper. Show that we can remove $7$ pieces so that the remaining $8$ cover at least $8/15$ of the table.

Each side face of a regular hexagonal prism is colored in one of three colors (for example, red, yellow, blue), and the adjacent prism faces have different colors. In how many different ways can the edges of the prism be colored (using all three colors is optional)?

6. The baker has baked a rectangular pancake. He then cut it into $n^{2}$ rectangles by making $n-1$ horizontal and $n-1$ vertical cuts. Being rounded to the closest integer, the areas of resulting rectangles equal to all positive integers from 1 to $n^{2}$ in some order. For which maximal $n$ could this happen? (Half-integers are rounded upwards.) Georgy Karavaev

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point $M$ outside it? (The polyhedron is made out of an opaque material.) (AY Belov, S Markelov)

Let $n$ be a positive integer. Bolek draws $2n$ points in the plane, no two of them defining a vertical or a horizontal line. Then Lolek draws for each of these $2n$ points two rays emanating from them, one of them vertically and the other one horizontally. Lolek wants to maximize the number of regions in which these rays divide the plane. Determine the largest number $k$ such that Lolek can obtain at least $k$ regions independent of the points chosen by Bolek.

A [i]form [/i] is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An [i]angle [/i] in a [i]form [/i] consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a [i]form [/i] of an [i]angle [/i] is the number of boxes it contains. Find the maximum number of [i]angles [/i] of size $11$ in a form of size $400$. [url=http://www.oma.org.ar/enunciados/omr20.htm]source[/url]

There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.

A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.

Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.

There are two kinds of creatures living in the flatland of Pentagonia: the Spires ($S$) and the Bones ($B$). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$. Every year on [i]Great Day of Division[/i] (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$'s and a $B$; each $B$ in an $S$ and a $B$. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$-being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

$n$ natural numbers ($n>3$) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number. Prove that the sum of those relations is a) not less than $2n$ b) less than $3n$

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