Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$

Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$

Is it possible in a plane mark $10$ red, $10$ blue and $10$ green points (all distinct) such that three conditions hold: $i)$ For every red point $A$ there exists a blue point closer to point $A$ than any other green point $ii)$ For every blue point $B$ there exists a green point closer to point $B$ than any other red point $iii)$ For every green point $C$ there exists a red point closer to point $C$ than any other blue point

Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove: $a)$ Triangle $AMD$ is isosceles $b)$ Circumcenter of $AMD$ lies on circle $C$

In triangle $ABC$ such that $\angle ACB=90^{\circ}$, let point $H$ be foot of perpendicular from point $C$ to side $AB$. Show that sum of radiuses of incircles of $ABC$, $BCH$ and $ACH$ is $CH$

Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$$$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$has solution in set of real numbers

Decomposition of number $n$ is showing $n$ as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer $n$ show that number of decompositions is $2^{n-1}$

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

In triangle $ABC$ holds $\angle ACB = 90^{\circ}$, $\angle BAC = 30^{\circ}$ and $BC=1$. In triangle $ABC$ is inscribed equilateral triangle (every side of a triangle $ABC$ contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board. $a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board $b)$ For such $n$, find numbers written on the board at the beginning

What is the minimal value of $\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}$, if $x$, $y$ and $z$ are nonnegative real numbers such that $x+y+z=4$

Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers

Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant