There is secret society with $2011$ members. Every member has bank account with integer balance ( can be negative). Sometimes some member give one dollar to every his friend. It is known, that after some such moves members can redistribute their money arbitrarily. Prove, that there are exactly $2010$ pairs of friends.

Let $n$ be a positive integer. We are given a regular $4n$-gon in the plane. We divide its vertices in $2n$ pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?

Saskia and her sisters have been given a large number of pearls. The pearls are white, black and red, not necessarily the same number of each color. Each white pearl is worth $5$ Ducates, each black one is worth $7$, and each red one is worth $12$. The total worth of the pearls is $2107$ Ducates. Saskia and her sisters split the pearls so that each of them gets the same number of pearls and the same total worth, but the color distribution may vary among the sisters. Interestingly enough, the total worth in Ducates that each of the sisters holds equals the total number of pearls split between the sisters. Saskia is particularly fond of the red pearls, and therefore makes sure that she has as many of those as possible. How many pearls of each color has Saskia?

Given positive integer $n,k$ such that $2 \le n <2^k$. Prove that there exist a subset $A$ of $\{0,1,\cdots,n\}$ such that for any $x \neq y \in A$, ${y\choose x}$ is even, and $$|A| \ge \frac{{k\choose \lfloor \frac{k}{2} \rfloor}}{2^k} \cdot (n+1)$$

Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?

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 $n$.

Each vertex of a regular $1997$-gon is labeled with an integer, so that the sum of the integers is $1$. We write down the sums of the first $k$ integers read counterclockwise, starting from some vertex $(k=1,2,\ldots,1997)$. Can we always choose the starting vertex so that all these sums are positive? If yes, how many possible choices are there?

For which $n \ge 1$ is it possible to place the numbers $1, 2, \ldots, n$ in some order $(a)$ on a line segment, or $(b)$ on a circle so that for every $s$ from $1$ to $\frac{n(n+1)}{2}$, there is a connected subset of the segement or circle such that the sum of the numbers in that subset is $s$?

In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to $ 1$, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to $ 1;$ A jump game is played by two frogs $ A,B,$ "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of $ A,B$" is called if first $ A$ jumps and then $ B$ by the following rules: Rule (1): $ A$ jumps once arbitrarily, then $ B$ jumps once in the same direction, or twice in the opposite direction; Rule (2): when $ A,B$ sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or $ A$ jumps twice continually, keep adjacent with $ B$ every time, and $ B$ rests on previous position; If the original positions of $ A,B$ are adjacent lattice points, determine whether for $ A$ and $ B$,such that the one can exactly land on the original position of the other after a finite round jumps.

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

The distance between Maykop and Belorechensk is $24$ km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most $6$ kmph) or on a bike (with velocity at most $18$ kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after $2$ hours $40$ minutes. (Only one guy may seat on the bike simultaneously). [i]Folclore[/i]

There are \(n+1\) squares in a row, labelled from 0 to \(n\). Tony starts with \(k\) stones on square 0. On each move, he may choose a stone and advance the stone up to \(m\) squares where \(m\) is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square \(n\). Show that Tony cannot achieve his goal in fewer than \(\frac{n}{1} + \frac{n}{2} + \cdots + \frac{n}{k}\) moves. [i]Proposed by Tony Wang[/i]

On a rectangular board $100 \times 300$, two people take turns coloring the cells that have not yet been colored. The first one colors cells in yellow, and the second one in blue. Coloring is completed when every cell of the board is colored. A [i]connected sequence[/i] of cells is a sequence of cells in which every two consecutive cells share a common side (and all cells in the sequence are different). Consider all possible connected sequences of yellow cells. The result of the first player is the number of cells in the connected sequence of yellow cells of maximum length. The first player's goal is to maximize the result, and the second player's goal is to make the first player's result as small as possible. Prove that if each player tries to achieve his goal, the result of the first player will be no more than $200$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]

$120$ bags with $100$ coins are placed on the floor. One bag has coins that weigh $9$ grams, the other bags have coins that weigh $10$ grams. One may place some coins (not necessarily from the same bag) on a weighing scale, but it will only properly display the weight if it is less than $1000$ grams. Determine the minimum number of times that the weighing scale may be used in order to identify the bag that has the $9$-gram coins.

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$. Find the maximum number of moves in a sequence for all possible initial sets.

Consider the sets $A_1,A_2,\dots,A_n$. Set $A_k$ is composed of $k$ disjoint intervals on the real axis ($k=1,2,\dots,n$). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$, and no two of these intervals have a common point.

The numbers from 1 to 37 are written in a line so that the sum of any first several numbers is divided by the number following them. What number is worth in third place, if the number 37 is written in the first place, and in the second, 1?

Grid the plane forming an infinite board. In each cell of this board, there is a lamp, initially turned off. A permitted operation consists of selecting a square of \(3\times 3\), \(4\times 4\), or \(5\times 5\) cells and changing the state of all lamps in that square (those that are off become on, and those that are on become off). (a) Prove that for any finite set of lamps, it is possible to achieve, through a finite sequence of permitted operations, that those are the only lamps turned on on the board. (b) Prove that if in a sequence of permitted operations only two out of the three square sizes are used, then it is impossible to achieve that at the end the only lamps turned on on the board are those in a \(2\times 2\) square.

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied: 1) there is at least $1$ marked point on each side, 2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$

Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things: 1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa) 2. Change the first bit after the first $1$. Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

There are 16 boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a 1$0$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.