Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]

A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$. Proposed by Michał Pilipczuk

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

A regular $n{}$-gon and a regular $m$-gon with distinct vertices are inscribed in the same circle. The vertices of these polygons divide the circle into $n+m$ arcs. Is it always possible to inscribe a regular $(n+m)$-gon in the same circle so that exactly one of its vertices is on each of these arcs?

For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Prove that for any points $A,B,C,D$ in the plane, the following inequality holds \[\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.\]

There are 100 saucers in a circle. Two people take turns putting marmalade of various colors in empty saucers. The first person can choose one or three empty saucers and fill each of them with marmalade of arbitrary color. The second one can choose one empty saucer and fill it with marmalade of arbitrary color. There should not be two adjacent saucers with marmalade of the same color. The game ends when all the saucers are filled. The loser is the last player to introduce a new color of marmalade into the game. Who has a winning strategy?

Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.

Several people came to the congress, each of whom has a certain number of tattoos on both hands. There are $n{}$ types of tattoos, and each of the $n{}$ types is found on the hands of at least $k{}$ people. For which pairs $(n, k)$ is it always possible for each participant to raise one of their hands so that all $n{}$ types of tattoos are present on the raised hands?

$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \]for every $x, y \in \mathbb{Z}$.

The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]

Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.

Let $x,y,z{}$ be positive real numbers. Prove that \[(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.\]

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

An Olympiad has 99 tasks. Several participants of the Olympiad are standing in a circle. They all solved different sets of tasks. Any two participants standing side by side do not have a common solved problem, but have a common unsolved one. Prove that the number of participants in the circle does not exceed \[2^{99}-\binom{99}{50}.\]

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]

The circles $\omega_1$ and $\omega_2$ intersect at $K{}$ and $L{}$. The line $\ell$ touches the circles $\omega_1$ and $\omega_2$ at the points $X{}$ and $Y{}$, respectively. The point $K{}$ lies inside the triangle $XYL$. The line $XK$ intersects $\omega_2$ a second time at the point $Z{}$. Prove that $LY$ is the bisector of the angle $XLZ$.

In the cyclic quadrilateral $ABCD$, the diagonal $BD$ is divided in half by the diagonal $AC$. The points $E, F, G$ and $H{}$ are the midpoints of the sides $AB, BC, CD{}$ and $DA$ respectively. Let $P = AD \cap BC$ and $Q = AB \cap CD{}$. The bisectors of the angles $APC$ and $AQC$ intersect the segments $EG$ and $FH$ at the points $X{}$ and $Y{}$ respectively. Prove that $XY \parallel BD$.

Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$

A simple graph has $N{}$ vertices and less than $3(N-1)/2$ edges. Prove that its vertices can be divided into two non-empty groups so that each vertex has at most one neighbor in the group it doesn't belong to.

A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.

Prove that \[1+\frac{2^1}{1-2^1}+\frac{2^2}{(1-2^1)(1-2^2)}+\cdots+\frac{2^{2016}}{(1-2^1)\cdots(1-2^{2016})}>0.\]

The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

Prove that a function $f:\mathbb{R}_+\to\mathbb{R}$ satisfies \[f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)\]if and only if it satisfies $f(xy)=f(x)+f(y)$.

The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$.

The squares $ABCD$ and $AXYZ$ are given. It turns out that $CDXY$ is a cyclic quadrilateral inscribed in the circle $\Omega$, and the points $A, B$ and $Z{}$ lie inside this circle. Prove that either $AB = AX$ or $AC\perp{}XY$.

In a class, there are $n{}$ children of different heights. Denote by $A{}$ the number of ways to arrange them all in a row, numbered $1,2,\ldots,n$ from left to right, so that each person with an odd number is shorter than each of his neighbors. Let $B{}$ be the number of ways to organize $n-1$ badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that $A = B$.

Prove that any rational number can be represented as a product of four rational numbers whose sum is zero.