Divide a regular $8960$-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$?

A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.

The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence $$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.

There is an $n \times n$ table with $n -1$ cells containing ones and the remaining cells containing zeros. You can do this with the table the following operation: select the tap hole, subtract from the number in this cell, one, and to all other numbers on the same line or in the same column as the selected cell, add one. Is it possible from of this table, using the specified operations, obtain a table in which all numbers are equal?

Each of $2n + 1$ students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least $n$ other students. Show that there is a student who is friends with everyone else.

Ana chose some unit squares of a $50 \times 50$ board and placed a chip on each of them. Prove that Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.

In the Cartesian plane, let $C$ be a unit circle with center at origin $O$. For any point $Q$ in the plane distinct from $O$, define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$. Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$.

Let $m,n\in \mathbb{Z}_{\ge 1}$ and a rectangular board $m\times n$ sliced by parallel lines to the rectangle's sides into $mn$ unit squares. At moment $t=0$, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn $180^o$ around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ [i](Oliver Hayman)[/i]

Zeroes are written in all cells of a $5\times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and the cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side.

Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements. Find $X$ such that the number of subsets with the same sum is maximum.

Each of the thieves in the $n$-member party ($n \ge 3$) charged a certain number of coins. All the coins were $100n$. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.

Given a positive integer $n$. A triangular array $(a_{i,j})$ of zeros and ones, where $i$ and $j$ run through the positive integers such that $i+j\leqslant n+1$ is called a [i]binary anti-Pascal $n$-triangle[/i] if $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$ for all possible values $i$ and $j$ may take on. Determine the minimum number of ones a binary anti-Pascal $n$-triangle may contain.

In each cell of a table $8\times 8$ lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible configurations are compatible with the statement.

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.

We are given five watches which can be winded forward. What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?

We say that a set $\displaystyle A$ of non-zero vectors from the plane has the property $\displaystyle \left( \mathcal S \right)$ iff it has at least three elements and for all $\displaystyle \overrightarrow u \in A$ there are $\displaystyle \overrightarrow v, \overrightarrow w \in A$ such that $\displaystyle \overrightarrow v \neq \overrightarrow w$ and $\displaystyle \overrightarrow u = \overrightarrow v + \overrightarrow w$. (a) Prove that for all $\displaystyle n \geq 6$ there is a set of $\displaystyle n$ non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$. (b) Prove that every finite set of non-zero vectors, which has the property $\displaystyle \left( \mathcal S \right)$, has at least $\displaystyle 6$ elements. [i]Mihai Baluna[/i]

Player Zero and Player One play a game on a $n \times n$ board ($n \ge 1$). The columns of this $n \times n$ board are numbered $1,2,4,\dots,2^{n-1}$. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a $0$ and Player One puts a $1$). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a $1$ in that row. For instance, when $n=4$, a row with $0101$ yields the number $0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$. a) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $4$? b) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $3$?

The game of pool includes $15$ balls that fit within a triangular rack as shown: [asy] // thanks Ritwin for this diagram :D unitsize(0.6cm); pair pos(real i, real j) { return i*dir(60) + (j,0); } for (int i = 0; i <= 4; ++i) { for (int j = 0; j <= 4-i; ++j) { draw(circle(pos(i,j), .5)); } } pair A = pos(0,0); pair B = pos(0,4); pair C = pos(4,0); pair dd = dir(270) * .5; pair ul = dir(150) * .5; pair ur = dir( 30) * .5; real S = 1.75; draw(A+dd -- B+dd ^^ B+ur -- C+ur ^^ C+ul -- A+ul ); draw(A+dd*S -- B+dd*S ^^ B+ur*S -- C+ur*S ^^ C+ul*S -- A+ul*S); draw(arc(A, A+ul*S, A+dd*S)); draw(arc(B, B+dd*S, B+ur*S)); draw(arc(C, C+ur*S, C+ul*S)); [/asy] Seven of the balls are "striped" (not colored with a single color) and eight are "solid" (colored with a single color). Prove that no matter how the $15$ balls are arranged in the rack, there must always be a pair of striped balls adjacent to each other.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.

Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time. Find the number of all attainable tables.

Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.

In each box of a $9\times 9$ grid we write a positive integer such that, between any $2$ boxes on the same row or column that have the same number $n$ written, there's at least $n$ boxes between them. What is the minimum sum possible for the numbers on the grid?