On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

Prove that in any convex polygon where all pairwise distances between vertices are distinct, there exists a vertex such that the closest vertex of the polygon is adjacent to it. [i]Proposed by D. Shiryayev, S. Berlov[/i]

There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than $100$ different ones. What is the largest possible value for $n$?

Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?

A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?

On an initially colourless plane three points are chosen and marked in red, blue and yellow. At each step two points marked in different colours are chosen. Then one more point is painted in the third colour so that these three points form a regular triangle with the vertices coloured clockwise in ''red, blue, yellow". A point already marked may be marked again so that it may have several colours. Prove that for any number of moves all the points containing the same colour lie on the same line.

* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?

Consider the part of a lattice given by the corners $(0, 0), (n, 0), (n, 2)$ and $(0, 2)$. From a lattice point $(a, b)$ one can move to $(a + 1, b)$ or to $(a + 1, b + 1)$ or to $(a, b - 1$), provided that the second point is also contained in the part of the lattice. How many ways are there to move from $(0, 0)$ to $(n, 2)$ considering these rules?

A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube. a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions. b) What is the maximum size of a regular subset that does exist for sure? (Nikita Gladkov, Alexandr Zimin)

There are $n$ red sticks and $n$ blue sticks. The sticks of each colour have the same total length, and can be used to construct an $n$-gon. We wish to repaint one stick of each colour in the other colour so that the sticks of each colour can still be used to construct an $n$-gon. Is this always possible if (a) $n = 3$, (b) $n > 3$ ?

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.

* What is the radius of the largest possible circle inscribed into a cube with side $a$?

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.

A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?

Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.

Five points are selected on the plane so that no three of them lie on one straight line. Prove that some four of these five points are the vertices of a convex quadrilateral.

Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.

Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs. (M. Kharitonov, A. Polyansky)

There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.

A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.

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.

Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices. [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]