Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

In the acute triangle $ABC$ point $M$ is the midpoint of $AC$ and $N$ is the foot of the height of $A$ on $BC$. Point $D$ is on the circumcircle of triangle $BMN$ such that $AD$ and $BM$ are parallel and $AC$ is between the points $B$ and $D$. Prove that $BD=BC$.

Find the smallest nonnegative integer $n$ such that in every set of $n$ numbers there are always two distinct numbers such that their sum or difference is divisible by $2022$.

Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

Solve the equation in $\mathbb{R}$ $$\left\{\left\{\frac{x^2-x}{2021}\right \}-\left\{\frac{x^2+x}{2022}\right \} \right \}=0.$$

Let $ABC$ be a triangle with $\angle ABC=130$. Point $D$ on side $AC$ is the foot of the perpendicular from $B$. Points $E$ and $F$ are on sides $(AB)$ and $(BC)$ such that $DE=DF$ and $AEFC$ is cyclic. Find $\angle EDF$.

Find all triplets of nonnegative integers $(x, y, z)$ that satisfy: $x^2-3y^2=y^2-3z^2=22$.

On a board there are $n\geq2$ distinct nonnegative integers such that the sum of each two distinct numbers is a power of $2$. What are the possible values of $n$?

There are $n\geq2$ numbers $x_1, x_2, \ldots, x_n$ such that $x^2_i=1 (1\leq i\leq n)$ and $$x_1x_2+x_2x_3+\dots+x_{n-1}x_n+x_nx_1=0.$$Prove that $n$ is divisible with $4$.

Compute $$\frac{\sum_{k=0}^{2022}\sin\frac{k\pi}{3033}}{\sum_{k=0}^{2022}\cos\frac{k\pi}{3033}}.$$

Let there be a trapezoid $ABCD$ with bases $AD$ and $BC$. Points $M$ and $P$ are on sides $AB$ and $CD$ such that $CM$ and $BP$ intersect in $N$ and the pentagon $AMNPD$ is cyclic. Prove that the triangle $ADN$ is isosceles.

On a board there are $2022$ numbers: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots,\frac{1}{2022}$. During a $move$ two numbers are chosen, $a$ and $b$, they are erased and $a+b+ab$ is written in their place. The moves take place until only one number is left on the board. What are the possible values of this number?