Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

Point $F$ lies on the circumscribed circle around $\Delta ABC$, $P$ and $Q$ are projections of point $F$ on $AB$ and $AC$ respectively. Prove that, if $M$ and $N$ are the middle points of $BC$ and $PQ$ respectively, then $MN$ is perpendicular to $FN$.

A row of $2n$ real numbers is called “Sozopolian”, if for each $m$, such that $1\leq m\leq 2n$, the sum of the first $m$ members of the row is an integer or the sum of the last $m$ members of the row is an integer. What’s the least number of integers that a Sozopolian row can have, if the number of its members is: a) 2016; b) 2017?

Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.

We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$ Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.

Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which $f(k+1)>f(f(k)) \quad \forall k \geq 1$.

There are 2017 points in a plane. For each pair of these points we mark the middle of the segment they form when connected. What’s the least number of marked points?

The points with integer coordinates in a plane are painted in two colors – blue and red. Prove that there exist an infinite monochromatic subset that is symmetrical on some point.

The function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called “Sozopolian”, if it satisfies the following two properties: For each two $x,y\in \mathbb{Z}$ which aren’t multiples of 17 the number $f(xy)-f(x)-f(y)$ is divisible by 8; For $\forall x\in \mathbb{Z}$ the number $f(x+17)-f(x)$ is divisible by 8. Does there exist a Sozopolian function for which a) $f(2)=1; \quad$ b) $f(3)=1$?

The lengths of the sides of a triangle are 19, 20, 21 cm. We can cut the triangle in a straight line into two parts. These two parts are put in a circle with radius $R$ cm without overlapping each other. Find the least possible value of $R$.

$n\in \mathbb{N}$ is called “good”, if $n$ can be presented as a sum of the fourth powers of five of its divisors (different). a) Prove that each good number is divisible by 5; b) Find a good number; c) Does there exist infinitely many good numbers?

Find all $n\in \mathbb{N}$, $n>1$ with the following property: All divisors of $n$ can be put in a rectangular table in such way that the sums of the numbers by rows are equal and the sums of the numbers by columns are also equal.

In a group of $n$ people $A_1,A_2… A_n$ each one has a different height. On each turn we can choose any three of them and figure out which one of them is the highest and which one is the shortest. What’s the least number of turns one has to make in order to arrange these people by height, if: a) $n=5$; b) $n=6$; c) $n=7$?

Find all triples $(x,y,z)$, $x,y,z\in \mathbb{Z}$ for which the number 2016 can be presented as $\frac{x^2+y^2+z^2}{xy+yz+zx}$.

The inscribed circle $\omega$ of an equilateral $\Delta ABC$ is tangent to its sides $AB$,$BC$ and $CA$ in points $D$,$E$, and $F$, respectively. Point $H$ is the foot of the altitude from $D$ to $EF$. Let $AH\cap BC=X,BH\cap CA=Y$. It is known that $XY\cap AB=T$. Let $D$ be the center of the circumscribed circle of $\Delta BYX$. Prove that $OH\perp CT$.

Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.

$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

With $\sigma (n)$ we denote the sum of the positive divisors of the natural number $n$. Prove that there exist infinitely many natural numbers $n$, for which $n$ divides $2^{\sigma (n)} -1$.

Let $n$ be a composite number and $a_1,a_2… a_k\in \mathbb{N}$ are the numbers smaller than $n$ and not coprime with it (in this case $k=n-\phi (n)$). Let $b_1,b_2…b_k$ be a permutation of $a_1,a_2… a_k$ Prove that there exist indexes $i$ and $j$, $i\neq j$ for which $a_i b_i\equiv a_j b_j (mod $ $n)$.

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

Let $p>5$ be a prime number. Prove that there exist $m,n\in \mathbb{N}$ for which $m+n<p$ and $2^m 3^n-1$ is a multiple of $p$.

The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.

Find all pairs $(x,y)$, $x,y\in \mathbb{N}$ for which $gcd(n(x!-xy-x-y+2)+2,n(x!-xy-x-y+3)+3)>1$ for $\forall$ $n\in \mathbb{N}$.

Let $\Delta ABC$ be a scalene triangle with center $I$ of its inscribed circle. Points $A_1$,$B_1$, and $C_1$ are the points of tangency of the same circle with $BC$,$CA$, and $AB$ respectively. Prove that the circumscribed circles of $\Delta AIA_1$,$\Delta BIB_1$, and $\Delta CIC_1$ intersect in a common point, different from $I$.

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.

Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.

$n$ students want to equally partition $m$ identical cakes between themselves. What’s the minimal number of pieces of cake one has to cut, so that the upper condition is satisfied? Each cut increases the number of pieces by 1.

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation $f(xy+x+y)=f(xy)+f(x)+f(y)$ is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, can't be covered with 2x1 domino tiles?

$k$ is the circumscribed circle of $\Delta ABC$. $M$ and $N$ are arbitrary points on sides $CA$ and $CB$, and $MN$ intersects $k$ in points $U$ and $V$. Prove that the middle points of $BM$,$AN$,$MN$, and $UV$ lie on one circle.