A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?

A cell is cut from a chessboard $8\, x\, 8$, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length $\sqrt{17}$ or $\sqrt{65}$. When is the number of such broken lines bigger – when the cut cell is $(1,2)$ or $(3,6)$? (The rows and columns on the board are numerated consecutively from 1 to 8.)

Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that: a) $x\leq 3$ and $z\geq 5$ b) $xy$, $yz$, $zx\in [9,25]$

Let $ABCD$ be a convex quadrilateral. The circle $\omega_1$ is tangent to $AB$ in $S$ and the continuations after $A$ and $B$ of sides $DA$ and $CB$, circle $\omega_2$ with center $I$ is tangent to $BC$ and the continuations after $B$ and $C$ of sides $AB$ and $DC$, circle $\omega_3$ is tangent to $CD$ in $T$ and the continuations after $C$ and $D$ of sides $BC$ and $AD$, and circle $\omega_4$ with center $J$ is tangent to $DA$ and the continuations after $D$ and $A$ of sides $CD$ and $BA$. Prove that points $S$ and $T$ are on equal distance from the middle point of segment $IJ$.

Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

Let $S$ be a set of integers which has the following properties: 1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$; 2) For $\forall$ $x,y\in S, x^2-y\in S$. Prove that $S\equiv \mathbb{Z}$ .

For a quadratic trinomial $f(x)$ and the different numbers $a$ and $b$ it is known that $f(a)=b$ and $f(b)=a$. We call such $a$ and $b$ conjugate for $f(x)$. Prove that $f(x)$ has no other conjugate numbers.

Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that: $\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.

Points $K$ and $L$ are inner for $AB$ for an acute $\Delta ABC$, where $K$ is between $A$ and $L$. Let $P,Q$, and $H$ be the feet of the perpendiculars from $A$ to $CK$, from $B$ to $CL$, and from $C$ to $AB$, respectively. Point $M$ is the middle point of $AB$. If $PH\cap AC=X$ and $QH\cap BC=Y$, prove that points $H,P,M$, and $Q$ lie on one circle, if and only if the lines $AY,BX$, and $CH$ intersect in one point.

We are given a chessboard 100 x 100, $k$ barriers (each with length 1), and one ball. We want to put the barriers between the cells of the board and put the ball in some cell, in such way that the ball can get to each possible cell on the board. The only way that the ball can move is by lifting the board so it can go only forward, backward, to the left or to the right. The ball passes all cells on its way until it reaches a barrier or the edge of the board where it stops. What’s the least number of barriers we need so we can achieve that?

A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\sqrt{98}$ and every odd one - $\sqrt{149}$. What’s the least number of jumps the grasshopper has to make in order to return to its starting point after odd number of jumps?

Find all triples of natural numbers $(x,y,z)$ for which: $xyz=x!+y^x+y^z+z!$.

The numbers from 1 to $n$ are arranged in some way on a circle. What’s the smallest value of $n$, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers $A_1<A_2<A_3…<A_{10}$ such that $A_1 A_2…A_{10}$ is a convex decagon?

Let $p$ be a prime number and the decimal notation of $\frac{1}{p}$ is periodical with a length of the period $4k$, $\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}…$ .Prove that $a_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}$.

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

We are given a $\Delta ABC$ with $\angle BAC=39^\circ$ and $\angle ABC=77^\circ$. Points $M$ and $N$ are chosen on $BC$ and $CA$ respectively, so that $\angle MAB=34^\circ$ and $\angle NBA=26^\circ$. Find $\angle BNM$.

$a,b,m,k\in \mathbb{Z}$, $a,b,m>2,k>1$, for which $k^n a+b$ is an $m$-th power of a natural number for $\forall n\in \mathbb{N}$. Prove that $b$ is an $m$-th power of a non-negative integer.

We are given a ruler with two marks at a distance 1. With its help we can do all possible constructions as with a ruler with no measurements, including one more: If there is a line $l$ and point $A$ on $l$, then we can construct points $P_1,P_2\in l$ for which $AP_1=AP_2=1$. By using this ruler, construct a perpendicular from a given point to a given line.

Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum $\sum_{i=1}^{2016}{a_i x_i^i}$ is a 2017-th power of a natural number.

There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.

A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides. a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle. b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.

Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?

We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.

Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions: 1) $f(x)\neq x$; 2) $f(f(x))=x$; 3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.

We are given a set $P$ of points and a set $L$ of straight lines. At the beginning there are 4 points, no three of which are collinear, and $L=\emptyset $. Two players are taking turns adding one or two lines to $L$, where each of these lines has to pass through at least two of the points in $P$. After that all intersection points of the lines in $L$ are added to $P$, if they are not already part of it. A player wins, if after his turn there are three collinear points from $P$, which lie on a line that isn’t from $L$. Find who of the two players has a winning strategy.

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.

$a$ and $b$ are fixed real numbers. With $x_n$ we denote the sum of the digits of $an+b$ in the decimal number system. Prove that the sequence $x_n$ contains an infinite constant subsequence.

Prove that for an arbitrary $\Delta ABC$ the following inequality holds: $\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?

Is the following set of prime numbers $p$ finite or infinite, where each $p$ doesn't divide the numbers that can be expressed as $n^{2016}+2016^{2016}$ for $n\in \mathbb{N}$, if: a) $p=4k+3$; b) $p=4k+1$?