Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.

$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied: If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$. Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation. (M Kontsevich, Moscow)

A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?

Compute the number of squares of positive area whose vertices all are points on the grid shown below. [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy]

Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. [hide=*]A tree is a connected graph with no cycles. A leaf is a vertex of degree 1[/hide] [i]Vincent Huang[/i]

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.

a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass. b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.

The plane is painted in red or blue color. Prove that you have a rectangle with the corners of the same color.

A family of sets $ A_1, A_2, \ldots ,A_n$ has the following properties: [b](i)[/b] Each $ A_i$ contains 30 elements. [b](ii)[/b] $ A_i \cap A_j$ contains exactly one element for all $ i, j, 1 \leq i < j \leq n.$ Determine the largest possible $ n$ if the intersection of all these sets is empty.

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.

$3n$ people forming $n$ families of a mother, a father and a child, stand in a circle. Every two neighbours can exchange places except the case when a parent exchanges places with his/her child (this is forbidden). For what $n$ is it possible to obtain every arrangement of those people by such exchanges? The arrangements differing by a circular shift are considered distinct.

Let $n> 2$ be a natural number. We consider $n$ candy bags, each containing exactly one candy. Ali and Omar play the following game in which they move alternately (Ali moves the first): At each move, the player who has to make a move chooses two bags containing $x$, respectively $y$ candy, with $(x,y)=1$, and he puts the $x + y$ candies in one bag (he chooses where). The player who can't make a move loses. Which of the two players has a strategy to win this game?

$n\ge 2$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. A team receives $2, 1, 0$ points for a win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings, the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings and ordered them by the total number of points. For which $n$ could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

In 11 cells of a square grid there live hedgehogs. Every hedgehog divides the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers? (V.Brayman)

There are $n \geq 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected to a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added. Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

There are $2025$ people and $66$ colors, where each person has one ball of each color. For each person, their $66$ balls have positive mass summing to one. Find the smallest constant $C$ such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed $C$.

Given is a group in which everyone has exactly $d$ friends and every two strangers have exactly one common friend. Prove that there are at most $d^2 + 1$ people in this group.

In an empty $100 \times 100$ grid, $300$ cells are colored blue, $3$ in each row and each column. Compute the largest positive integer $k$ such that you can always recolor $k$ of these blue cells red so that no contiguous $2 \times 2$ square has four red cells. [i]Arul Kolla[/i]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates (a) four evenings are not enough (b) five evenings are not enough (c) ten evenings are enough (d) even seven evenings are enough