Prove that for positive real numbers $a,b,c$ satisfying $abc=1$ the following inequality holds: $$ \frac{a}{b(1+c)} +\frac{b}{c(1+a)}+\frac{c}{a(1+b)} \ge \frac{3}{2} $$

Determine all functions $f: \mathbb{R} \backslash \{0 \} \rightarrow \mathbb{R}$ such that, for all nonzero $x$: $$ f(\frac{1}{x}) \ge 1 -f(x) \ge x^2f(x) $$

Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds: $$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.

Six lines are drawn in the plane. Determine the maximum number of points, through which at least $3$ lines pass.

Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.

$22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.

Initially on the blackboard eight zeros are written. In one step, it is allowed to choose numbers $a,b,c,d$, erase them and replace them with the numbers $a+1$, $b+2$, $c+3$, $d+3$. Determine: a) the minimum number of steps required to achieve $8$ consecutive integers on the board b) whether it is possible to achieve that sum of the numbers is $2021$ c) whether it is possible to achieve that product of the numbers is $2145$

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

Circle $\omega$ with centre $M$ and diameter $XY$ is given. Point $A$ is chosen on $\omega$ so that $AX<AY$. Points $B, C$ are chosen on segments $XM, YM$, respectively, in a way that $BM=CM$. A parallel line to $AB$ is constructed through $C$; the line intersects $\omega$ at $P$ so that $P$ lies on the smaller arc $\widehat{AY}$. Similarly, a parallel line to $AC$ is constructed through $B$; the line intersects $\omega$ at $Q$ so that $Q$ lies on the smaller arc $\widehat{XA}$. Lines $PQ$ and $XY$ intersect at $S$. Prove that $AS$ is tangent to $\omega$.

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.

Does there exist a natural number $a$ so that: a) $\Big ((a^2-3)^3+1\Big) ^a-1$ is a perfect square? b) $\Big ((a^2-3)^3+1\Big) ^{a+1}-1$ is a perfect square?

Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and $$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$

Denote by $s(n)$ the sum of all natural divisors of $n$ which are smaller than $n$. Does there exist a positive integer $a$ such that the equation $$s(n)=a+n$$has infinitely many solutions in positive integers?

A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$Find all possible values of $f(11)$.