Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

A bag contains $12$ marbles: $3$ red, $4$ green, and $5$ blue. Repeatedly draw marbles with replacement until you draw two marbles of the same color in a row. What is the expected number of times that you will draw a marble?

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

A professor organized five exams for a class consisting of at least two students. Before starting the first test, he deduced that there will be at least two students from that class that will have the same amount of passed exams. What is the minimum numer of students that class could have had such that the conclusion of the professor's reasoning was correct.

Two children take turns breaking chocolate bar that is 5*10 squares. They can only break the bar using the divisions between squares and can only do 1 break at a time.. The first player that when breaking the chocolate bar breaks off only a single square wins. Is there a winning strategy for any player?

Let $\Omega$ be a family of subsets of $M$ such that: $(\text{i})$ If $|A\cap B|\ge 2$ for $A,B\in\Omega$, then $A=B$; $(\text{ii})$ There exist different subsets $A,B,C\in\Omega$ with $|A\cap B\cap C|=1$; $(\text{iii})$ For every $A\in\Omega$ and $a\in M \ A$, there is a unique $B\in\Omega$ such that $a\in B$ and $A\cap B=\emptyset$. Prove that there are numbers $p$ and $s$ such that: $(1)$ Each $a\in M$ is contained in exactly $p$ sets in $\Omega$; $(2)$ $|A|=s$ for all $A\in\Omega$; $(3)$ $s+1\ge p$.

A segment $AB$ of length $100$ is divided into $100$ little segments of length $1$ by $99$ intermediate points. Endpoint $A$ is assigned $0$ and endpoint $B$ is assigned $1$. Gustavo assigns each of the $99$ intermediate points a $0$ or a $1$, at his choice, and then color each segment of length $1$ blue or red, respecting the following rule: [i]The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. [/i] Determine if Gustavo can assign the $0$'s and $1$'s so as to get exactly $30$ blue segments. And $35$ blue segments? (In each case, if the answer is yes, show a distribution of $0$'s and $1$'s, and if the answer is no, explain why).

For odd integers $n,$ two people play the game on $2\times n$ grid. Each people color one cell that is not colored before with white and black. When coloring is done, they count the number of ordered pairs of neighboring cells that have the same color and different color, respectively. If same color neighboring ordered pair of cells are more than different color neighboring ordered pair of cells, the person who first starts win and lose otherwise. (If the number is same, they are tied.) If both of them use the best strategy, find the result of the game.

Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible a) for $k=3$, b) for every $k$ less than $(n-1)$.

In a group of people, every two of them have exactly one mutual friend in the group. Prove that there is one person who is friends with all the other people in the group. Note: the friendship is mutual, that is, if $X$ is friends with $Y$, then $Y$ is friends with $X$.

$n$ cockroaches are sitting on the plane at the vertices of the regular $ n $ -gon. They simultaneously begin to move at a speed of $ v $ on the sides of the polygon in the direction of the clockwise adjacent cockroach, then they continue moving in the initial direction with the initial speed. Vasya a entomologist moves on a straight line in the plane at a speed of $u$. After some time, it turned out that Vasya has crushed three cockroaches. Prove that $ v = u $.

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

Merlin's castle has 100 rooms and 1000 corridors. Each corridor links some two rooms. Each pair of rooms is linked by one corridor at most. Merlin has given out the plan of the castle to the wise men and declared the rules of the challenge. The wise men need to scatter across the rooms in a manner they wish. Each minute Merlin will choose a corridor and one of the wise men will have to pass along it from one of the rooms at its ends to the other one. Merlin wins when in both rooms on the ends of the chosen corridor there are no wise men. Let us call a number $m$ the magic number of the castle if $m$ wise men can pre-agree before the challenge and act in such a way that Merlin never wins, $m$ being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men). Timofey Vasilyev

There are $2010$ cities in country, and $3$ roads go from every city. President and Prime Minister play next game. They sell roads by turn to one of $3$ companies( one road is one turn). President will win, if three roads from some city are sold to different companies. Who will win?

Let $n\geq 2$ be an even integer. Find the greatest integer $m\geq 2^{n-2}+1$ such that there exist $m$ distinct subsets of $\{1,2,\dots ,n\}$, any $2^{n-2}+1$ of them having empty intersection. [i]Cristi Săvescu[/i]

Let $n$ be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of $1,2$ or $3$ in every cell, which simultaneously satisfies the following conditions: (1) Any two cells which share a common side does not have the same number filled in them. (2) For any $1\times 3$ or $3\times 1$ subgrid, the numbers filled does not contain $1,2,3$ in that order be it reading from top to bottom, bottom to top, or left to right, or right to left. (3) The sum of numbers of any $n\times n$ subgrid is the same.

The edges of a complete graph on $1000$ vertices are colored in three colors. Prove that this graph contains a non-self-intersecting single-color cycle whose length is odd and not less than $41$.

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares [b]2 x 2[/b]. Prove that the dominoes in a valid covering of the table are at least $\frac{3}{2}(x+y)-6$

Let $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that $\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and $\bullet$ each point ($x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$. During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number. Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.