Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

In the interior of a circle centred at $O$ consider the $1200$ points $A_1,A_2,\ldots ,A_{1200}$, where for every $i,j$ with $1\le i\le j\le 1200$, the points $O,A_i$ and $A_j$ are not collinear. Prove that there exist the points $M$ and $N$ on the circle, with $\angle MON=30^{\circ}$, such that in the interior of the angle $\angle MON$ lie exactly $100$ points.

Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!

Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.

Let $A$ be a non-empty subset of $\mathbb{R}$ with the property that for every real numbers $x,y$, if $x+y\in A$ then $xy\in A$. Prove that $A=\mathbb{R}$.

Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K,L,M,N$ on the lines $DA,AB,BC,CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.

Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.

Let $n$ be a non-negative integer. Find all non-negative integers $a,b,c,d$ such that \[a^2+b^2+c^2+d^2=7\cdot 4^n\]

Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.

Let $n\ge 2$ be a positive integer. Find the positive integers $x$ \[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \] for any number of radicals.

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.