In a box there are $31$, $41$ and $59$ stones coloured, respectively, red, green and blue. Three players, having t-shirts of these three colours, play the following game. They sequentially make one of two moves: (I) either remove three stones of one colour from the box, (II) or replace two stones of different colours by two stones of the third colour. The game ends when all the stones in the box have the same colour and the winner is the player whose t-shirt has this colour. Assuming that the players play optimally, is it possible to decide whether the game ends and who will win, depending on who the starting player is?

In a $k$-player tournament for $k > 1$, every player plays every other player exactly once. Find with proof the smallest value of $k$ such that it is possible that for any two players, there was a third player who beat both of them.

Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ? (Assume no number starts with zero.)

A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals. For every vertex the number of sides issuing from it is even, except for the vertices $A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and \[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\] if $k>0$ and \[n\equiv0\pmod3\mbox{ for }k=0.\] Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.

Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]: [list] [*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order) [*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order) [/list] What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves? [i]Proposed by Milan Haiman[/i]

Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions. (1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.) (2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$? (3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns. The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.

Let $n\geq 2$ be a positive integer. The sets $A_{1},A_{2},\ldots, A_{n}$ and $B_{1},B_{2},\ldots, B_{n}$ of positive integers are such that $A_{i}\cap B_{j}$ is non-empty $\forall i,j\in\{1,2,\ldots ,n\}$ and $A_{i}\cap A_{j}=\o$, $B_{i}\cap B_{j}=\o$ $\forall i\neq j\in \{1,2,\ldots, n\}$. We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Suppose $F$ is a family of finite subsets of $\mathbb{N}$ and for any 2 sets $A,B \in F$ we have $A \cap B \not= \O$. (a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A,B \in F$ we have $A\cap B\cap Y \not= \O$? (b) Is the above true if we assume that all members of $F$ have the same size?

The sequence $a_1,a_2,\dots$ of integers satisfies the conditions: (i) $1\le a_j\le2015$ for all $j\ge1$, (ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$. Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$. [i]Proposed by Ivan Guo and Ross Atkins, Australia[/i]

For a sequence $a_1,a_2,...,a_m$ of real numbers, define the following sets \[A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\] Let $n$ be a given integer, and $n>2$. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set $A\triangle B$, where $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$

In a mathematical challenge, positive real numbers $a_{1}\geq a_{2} \geq ... \geq a_{n}$ and an initial sequence of positive real numbers $(b_{1}, b_{2},...,b_{n+1})$ are given to Secco. Let $C$ a non-negative real number. In a sequence $(x_{1},x_{2},...,x_{n+1})$, consider the following operation: Subtract $1$ of some $x_{j}$, $j \in \{1,2,...,n+1\}$, add $C$ to $x_{n+1}$ and replace $(x_{1},x_{2},...,x_{j-1})$ for $(x_{1}+a_{\sigma (1)}, x_{2}+a_{\sigma (2)}, ..., x_{j-1}+a_{\sigma (j-1)})$, where $\sigma$ is a permutation of $(1,2,...,j-1)$. Secco's goal is to make all terms of sequence $(b_{k})$ negative after a finite number of operations. Find all values of $C$, depending of $a_{1}, a_{2},..., a_{n}, b_{1}, b_{2}, ..., b_{n+1}$, for which Secco can attain his goal.

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist. [i]Proposed by Toomas Krips, Estonia[/i]

The numbers from $1$ to $n^2$ are written in order in a grid of $n \times n$, one number in each square, in such a way that the first row contains the numbers from $1$ to $n$ from left to right; the second row contains the numbers $n + 1$ to $2n$ from left to right, and so on and so forth. An allowed move on the grid consists in choosing any two adjacent squares (i.e. two squares that share a side), and add (or subtract) the same integer to both of the numbers that appear on those squares. Find all values of $n$ for which it is possible to make every squares to display $0$ after making any number of moves as necessary and, for those cases in which it is possible, find the minimum number of moves that are necessary to do this.

Let $n$ points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?

Show that any graph with $6$ points has a triangle or three points which are not joined to each other.

Of six cities $S_1,S_2,...,S_6$ and two airlines $A$ and $B$ it is given that for every pair $(S_i,S_j)$ (where $i \ne j$) exactly one of the airlines has a connection from $S_i$ to $S_j$ and maintains back. (a) Prove that the air net of one of the companies contains a triangle. (b) Prove that in the two air nets there are even two triangles. [hide=]original wording]Van zes steden $S_1,S_2,...,S_6$ en twee luchtvaartmaatschappijen $A$ en $B$ is gegeven, dat voor ieder paar $(S_i,S_j)$ (waar $i \ne j$) precies één van de maatschappijen een verbinding van $S_i$ naar $S_j$ en terug onderhoudt, (a) Bewijs, dat het luchtnet van één van de maaschappijen een driehoek bevat; (b) Bewijs, dat er in de twee luchtnetten zelfs twee driehoeken zijn.[/hide]

Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal. Proposed by [i]Sena Başaran[/i]

Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number [list][b]a)[/b] $x^2$; [b]b)[/b] $xy$?[/list]

Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?