Show that for every integer $n \geq 2$ the number $$a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1$$is a composite number.

Real numbers $a, b, c, d$ satisfy $$a^2+b^2+c^2+d^2=4.$$Find the greatest possible value of $$E(a,b,c,d)=a^4+b^4+c^4+d^4+4(a+b+c+d)^2 .$$

Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

In the acute triangle $ABC$ the point $M$ is on the side $BC$. The inscribed circle of triangle $ABM$ touches the sides $BM$, $MA$ and $AB$ in points $D$, $E$ and $F$, and the inscribed circle of triangle $ACM$ touches the sides $CM$, $MA$ and $AC$ in points $X$, $Y$ and $Z$. The lines $FD$ and $ZX$ intersect in point $H$. Prove that lines $AH$, $XY$ and $DE$ are concurrent.

The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies: $1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$ $2) f(20^{22})=f(22^{20});$ $3) f(2021)=2022$. Find all possible values of $f(2022)$.

Let $A$ be a point outside of the circle $\Omega$. Tangents from $A$ touch $\Omega$ in points $B$ and $C$. Point $C$, collinear with $A$ and $P$, is between $A$ and $P$, such that the circumcircle of triangle $ABP$ intersects $\Omega$ again in point $E$. Point $Q$ is on the segment $BP$, such that $\angle PEQ=2 \cdot \angle APB$. Prove that the lines $BP$ and $CQ$ are perpendicular.

Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible.

a) Let $n$ $(n \geq 2)$ be an integer. On a line there are $n$ distinct (pairwise distinct) sets of points, such that for every integer $k$ $(1 \leq k \leq n)$ the union of every $k$ sets contains exactly $k+1$ points. Show that there is always a point that belongs to every set. b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer $k$ the union of every $k$ sets contains exactly $k+1$ points?

Let $n$ be a positive integer. A grid of dimensions $n \times n$ is divided in $n^2$ $1 \times 1$ squares. Every segment of length $1$ (side of a square) from this grid is coloured in blue or red. The number of red segments is not greater than $n^2$. Find all positive integers $n$, for which the grid always will cointain at least one $1 \times 1$ square which has at least three blue sides.

Let $P(X)$ be a polynomial with positive coefficients. Show that for every integer $n \geq 2$ and every $n$ positive numbers $x_1, x_2,..., x_n$ the following inequality is true: $$P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.$$When does the equality take place?

Let $\Omega$ be the circumcircle of triangle $ABC$ such that the tangents to $\Omega$ in points $B$ and $C$ intersect in $P$. The squares $ABB_1B_2$ and $ACC_1C_2$ are constructed on the sides $AB$ and $AC$ in the exterior of triangle $ABC$, such that the lines $B_1B_2$ and $C_1C_2$ intersect in point $Q$. Prove that $P$, $A$, and $Q$ are collinear.

Let $(x_n)_{n\geq1}$ be a sequence that verifies: $$x_1=1, \quad x_2=7, \quad x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2.$$Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$