Let there be $2004$ be bicolor tiles, white on one side and black on the other, placed in a circle. A move consists of choosing a black piece and turning over three pieces: the chosen one, the one on its left and the one on its right. If at the beginning there is only one black piece, will it be possible, repeating the movement described, to make all the pieces have the white face up?

Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.

On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible.

For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called [i]complete[/i] if the following conditions are met: i. In each cell is written either a "$0$", a "$1$" or nothing; ii. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied). A [i]complete[/i] rectangle is called [i]minimal[/i], if we remove any of its rows and then making it no longer [i]complete[/i]. a. Prove that for any positive integer $k\le 2018$ there exists a [i]minimal[/i] $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$. b. A [i]minimal[/i] $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$.

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

Let $ n \geq 2$ be an integer, let $ S$ be a set of $ n$ elements, and let $ A_i , \; 1\leq i \leq m$, be distinct subsets of $ S$ of size at least $ 2$ such that \[ A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ .\] Show that $ m \leq 2^{n\minus{}1}\minus{}1$. [i]P. Erdos[/i]

For positive integers $t,a,b,$a $(t,a,b)$-[i]game[/i] is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.

Let $E$ denote the set of $25$ points $(m,n)$ in the $\text{xy}$-plane, where $m,n$ are natural numbers, $1\leq m\leq5,1\leq n\leq5$. Suppose the points of $E$ are arbitrarily coloured using two colours, red and blue. SHow that there always exist four points in the set $E$ of the form $(a,b),(a+k,b),(a+k,b+k),(a,b+k)$ for some positive integer $k$ such that at least three of these four points have the same colour. (That is, there always exist four points in the set $E$ which form the vertices of a square with sides parallel to the axes and having at least three points of the same colour.)

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

Let $T$ be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points $(x, y, z)$ and $(u, v,w)$ are called [i]neighbors[/i] if $|x - u| + |y - v| + |z - w| = 1$. Show that there exists a subset $S$ of $T$ such that for each $p \in T$ , there is exactly one point of $S$ among $p$ and its [i]neighbors[/i].

Katie has a fair $2019$-sided die with sides labeled $1, 2,..., 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{th}$ roll is a $ 2019$?

A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$ [i]CSJL[/i]

Kelvin the frog is on the bottom-left lily pad of a $3 \times 3$ grid of lily pads, and his home is at the top right lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the bottom-left and top-right lily pads both remain, but Kelvin cannot get home.

A lattice point $(x, y, z) \in Z^3$ can be seen from the origin if the line from the origin does not contain any other lattice point $(x', y', z')$ with $$(x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2.$$ Let $p$ be the probability that a randomly selected point on the cubic lattice $Z^3$ can be seen from the origin. Given that $$\frac{1}{p}= \sum^{\infty}_{n=i} \frac{k}{n^s}$$ for some integers $ i, k$, and $s$, find $i, k$ and $s$.

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

In Shvambrania there are $N$ towns, every two of which are connected by a road. These roads do not intersect. If necessary, some of them pass over or under others via bridges. An evil magician establishes one-way rules along the roads in such a way that if someone goes out of a certain town he is unable to come back. Prove that (a) It is possible to establish such rules. (b) There exists a town from which it is possible to reach any other town, and there exists a town from which it is not possible to go out. (c) There is one and only one route passing through all towns. (d) The magician can realise his intention in $N!$ ways. (LM Koganov, Moscow) PS. (a),(b),(c) for Juniors, (a),(b),(d) for Seniors

Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser? [img]http://i38.tinypic.com/2i1icec_th.png[/img] [img]http://i34.tinypic.com/28rk4s3_th.png[/img] a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime. b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser. Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.

Let ${A}$ be a finite set and ${\rightarrow}$ be a binary relation on it such that for any ${a,b,c \in A}$, if ${a\neq b}, {a \rightarrow c}$ and ${b \rightarrow c}$ then either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Let ${B,\,B \subset A}$ be minimal with the property: for any ${a \in A \setminus B}$ there exists ${b \in B}$, such that either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Supposing that ${A}$ has at most ${k}$ elements that are pairwise not in relation ${\rightarrow}$, prove that ${B}$ has at most ${k}$ elements.

A square of sidelength $2^n$ is divided into unit squares. One of the unit squares is deleted. Prove that the rest of the square can be tiled with $L$-trominos.

Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?

The cells of a $50\times 50$ square are painted in four colors. Prove that there is a cell, on four sides of which (i.e. top, bottom, left and on the right) there are cells of the same color as it (not necessarily adjacent to this cell).

The set of positive integers is partitionated in subsets with infinite elements each. The question (in each of the following cases) is if there exists a subset in the partition such that any positive integer has a multiple in this subset. a) Prove that if the number of subsets in the partition is finite the answer is yes. b) Prove that if the number of subsets in the partition is infinite, then the answer can be no (for a certain partition).