Arne and Berit are playing a game. They have chosen positive integers $m$ and $n$ with $n\geq 4$ and $m \leq 2n + 1$. Arne begins by choosing a number from the set $\{1, 2, \dots , n \}$, and writes it on a blackboard. Then Berit picks another number from the same set, and writes it on the board. They continue alternating turns, always choosing numbers that are not already on the blackboard. When the sum of all the numbers on the board exceeds or equals $m$, the game is over, and whoever wrote the last number has won. For which combinations of $m$ and $n$ does Arne have a winning strategy?

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations - to take 2 numbers from these, to double first one, to add the second one and to save the sum. - to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. - to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.)

What condition should the angles of triangle $ ABC $ satisfy so that the bisector of angle $ A $, the median drawn from vertex $ B $ and the altitude drawn from vertex $ C $ intersect at one point?

Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.

Given an acute angle triangle $ABC$ with circumcircle $\Gamma$ and circumcenter $O$. A point $P$ is taken on the line $BC$ but not on $[BC]$. Let $K$ be the reflection of the second intersection of the line $AP$ and $\Gamma$ with respect to $OP$. If $M$ is the intersection of the lines $AK$ and $OP$, prove that $\angle OMB+\angle OMC=180^{\circ}$.

Let $\triangle ABC$ be an acute triangle, and let $D,E,F$ respectively be three points on sides $BC,CA,AB$ such that $AEDF$ is a cyclic quadrilateral. Let $O_B$ and $O_C$ be the circumcenters of $\triangle BDF$ and $\triangle CDE$, respectively. Finally, let $D'$ be a point on segment $BC$ such that $BD'=CD$. Prove that $\triangle BD'O_B$ and $\triangle CD'O_C$ have the same surface.

Given a positive integer $c$, we construct a sequence of fractions $a_1, a_2, a_3,...$ as follows: $\bullet$ $a_1 =\frac{c}{c+1} $ $\bullet$ to get $a_n$, we take $a_{n-1}$ (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add $2$ to the numerator and $3$ to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator. For example, if we take $c = 20$, then $a_1 =\frac{20}{21}$ and $a_2 =\frac{22}{24} = \frac{11}{12}$ . Then we find that $a_3 =\frac{13}{15}$ (which is already simplified) and $a_4 =\frac{15}{18} =\frac{5}{6}$. (a) Let $c = 10$, hence $a_1 =\frac{10}{11}$ . Determine the largest $n$ for which a simplification is needed in the construction of $a_n$. (b) Let $c = 99$, hence $a_1 =\frac{99}{100}$ . Determine whether a simplification is needed somewhere in the sequence. (c) Find two values of $c$ for which in the first step of the construction of $a_5$ (before simplification) the numerator and denominator are divisible by $5$.

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal. [i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.[/i]

Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;

The leader of the Gnome country wants to print banknotes in $12$ different denominations (each with an integer number) in such a way that it is possible to pay an arbitrary amount from $1$ to $6543$ with these banknotes without a balance, using a maximum of $8$ banknotes. (Several bills with the same denomination can be used during payment.) Can the leader of the land of Gnomes do it?

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

[b]p1.[/b] On the island of Nevermind some people are liars; they always lie. The remaining habitants of the island are truthlovers; they tell only the truth. Three habitants of the island, $A, B$, and $C$ met this morning. $A$ said: “All of us are liars”. $B$ said: “Only one of us is a truthlover”. Who of them is a liar and who of them is a truthlover? [b]p2.[/b] Pinocchio has $9$ pieces of paper. He is allowed to take a piece of paper and cut it in $5$ pieces or $7$ pieces which increases the number of his pieces. Then he can take again one of his pieces of paper and cut it in $5$ pieces or $7$ pieces. He can do this again and again as many times as he wishes. Can he get $2004$ pieces of paper? [b]p3.[/b] In Dragonland there are coins of $1$ cent, $2$ cents, $10$ cents, $20$ cents, and $50$ cents. What is the largest amount of money one can have in coins, yet still not be able to make exactly $1$ dollar? [b]p4.[/b] Find all solutions $a, b, c, d, e$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & d \\ + & a & c & a & c \\ \hline c & d & e & b & c \\ \end{tabular}$ [b]p5.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In an acute triangle $\triangle ABC$, let there be point $D$ on segment $AC, E$ on segment $AB$ such that $\angle ADE = \angle ABC$. Let the bisector of $\angle A$ hit $BC$ at $K$. Let the foot of the perpendicular from $K$ to $DE$ be $P$, and the foot of the perpendicular from $A$ to $DE$ be $L$. Let $Q$ be the midpoint of $AL$. If the incenter of $\triangle ABC$ lies on the circumcircle of $\triangle ADE$, prove that $P,Q$ and the incenter of $\triangle ADE$ are collinear.

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Let $a,b,c \in \mathbb{R}^+$ and $abc = 1$ such that $a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}$. Prove that $a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}$.

For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?

Find, with proof, all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$ has a solution in positive integers.

Let $a,b,c$ be real numbers which satisfy: $$a+b+c=2$$ $$a^2+b^2+c^2=2$$ Prove that at least one of numbers $|a-b|, |b-c|, |c-a|$ is greater or equal than $1$.

Let $a,b,c,d$ real numbers such that $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximum value of \[ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} \]

Let $C$ and $D$ be two distinct points on a semicircle of diameter $AB$. Let $E$ be the intersection of $AC$ and $BD$, $F$ be the intersection of $AD$ and $BC$ and $X, Y$, and $Z$ are the midpoints of $AB, CD$, and $EF$, respectively. Prove that the points $X, Y,$ and $Z$ are collinear.

Suppose that $a_{ij}$ are real numbers in such a way that for each $i$, the series $\sum_{j=1}^{\infty}a_{ij}$ is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence $\{b_j\}_j$ we have $\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0$. Prove that $\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0$.

Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that \[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]

Is it possible to color points in the Cartesian Plane $(x,y)$ with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the $x$-axis and that any three points, each of a different color, are not in a line? Justify your answer.

Let $x_1, x_2, ..., x_{24}$ be real numbers. prove that $$x_1 + 2x_2 + 3x_3 +...+ 24x_{24} - 439 \le \frac{x^2_1+x^2_2+... + x^2_{24}}{2}+ 2011.$$