Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90 degrees left turn after every $ \ell$ kilometer driving from start, Rob makes a 90 degrees right turn after every $ r$ kilometer driving from start, where $ \ell$ and $ r$ are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair ($ \ell$, $ r$) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

An ordered pair of sets $(A, B)$ is [i]good[/i] if $A$ is not a subset of $B$ and $B$ is not a subset of $A$. How many ordered pairs of subsets of $\{1, 2, \dots, 2017\}$ are good?

In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.

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.

Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper. Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation). [i]Proposed by Stephan Wagner, Austria[/i]

On a board with $3$ columns and $4$ rows, each of the $12$ squares will be painted green or white. In the first and last row, the number of squares painted green must be the same. Furthermore, in the first and last column, the number of squares painted green must also be unequal. How many different ways can you paint the board?

A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?

Two players by turns draw diagonals in a regular $(2n+1)$-gon ($n>1$). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy? [i]K. Sukhov[/i]

A collection $F$ of distinct (not necessarily non-empty) subsets of $X = \{1,2,\ldots,300\}$ is [i]lovely[/i] if for any three (not necessarily distinct) sets $A$, $B$ and $C$ in $F$ at most three out of the following eight sets are non-empty \begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C} \end{align*} where $\overline{S}$ denotes the set of all elements of $X$ which are not in $S$. What is the greatest possible number of sets in a lovely collection?

A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.

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].

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.

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]

A number is written on the board. Petya can change the number on the board to the sum of the squares of digits of the number on the board. A number is called interesting if Petya, when starting from this number, will not ever get the number on the board to be $1$. Prove that there infinitely many interesting numbers.

The first $360$ natural numbers are separated into $9$ blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining $9$ numbers. Is it possible to fill a $3 \times 3$ grid and form a [i]magic square[/i] with these numbers? Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.

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.

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies? [i](N. Chernyatevya)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

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.

It is considered a regular polygon with $n$ sides, where $n(n>3)$ is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen $m(0\leq m\leq n)$ vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered $monocolor$ ,if all of its vertices are of the same color. Prove that the number of all $monocolor$ isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these $monocolor$ isosceles triangles.