Three coins are placed at the origin of a Cartesian coordinate system. On one move one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$ and $(x+1, y+1)$. Prove that after finitely many moves, there will exist two coins placed at the same point.

Find all pairs of positive integers $(x, y)$, such that $x^3+9x^2-11x-11=2^y$.

Let $S$ be the set of all convex cyclic heptagons in the plane. Define a function $f:S \rightarrow \mathbb{R}^+$, such that for any convex cyclic heptagon $ABCDEFG,$ $$f(ABCDEFG)=\frac{AC \cdot BD \cdot CE \cdot DF \cdot EG \cdot FA \cdot GB} {AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FG \cdot GA}. $$ a) Show that for any $M \in S$, $f(M) \geq f(\prod)$, where $\prod$ is a regular heptagon. b) If $f(M)=f(\prod)$, is it true that $M$ is a regular heptagon?

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a bijection and let $k$ be a positive integer such that $|f(x+1)-f(x)| \leq k$ for all positive integers $x$. Show that there exists an integer $d$, such that $f(x)=x+d$ for infinitely many positive integers $x$.

The circles $k_1, k_2$, centered at $O_1, O_2$, meet at two points, one of which is $A$. Let $P, Q$ lie on $AO_1, AO_2$, respectively, so that $PQ \parallel O_1O_2$. The tangents from $P$ to $k_2$ touch it at $X, Y$ and the tangents from $Q$ to $k_1$ touch it at $Z, T$. Show that $X, Y, Z, T$ are collinear or concyclic.

In the plane, there is a figure in the form of an $L$-tromino, which is composed of $3$ unit squares, which we will denote by $\Phi_0$. On every move, we choose an arbitrary straight line in the plane and using it we construct a new figure. The $\Phi_n$, obtained in the $n$-th move, is obtained as the union of the figure $\Phi_{n-1}$ and its axial reflection with respect to the chosen line. Also, for the move to be valid, it is necessary that the surface of the newly obtained piece to be twice as large as the previous one. Is it possible to cover the whole plane in that process?

Does there exist a positive integer $n$ and a) complex numbers $a_0, a_1, \ldots, a_n;$ b) reals $a_0, a_1, \ldots, a_n, $ such that $P(x) Q(x)=x^{2024}+1$ where $P(x)=a_nx^n+\ldots +a_1x+a_0$ and $Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?$

Let $n$ be a positive integer. Initially a few positive integers are written on the blackboard. On one move Igor chooses two numbers $a, b$ of the same parity on the blackboard and writes $\frac{a+b} {2}$. After a few moves the numbers on the blackboard were exactly $1, 2, \ldots, n$. Find the smallest possible number of positive integers that were initially written on the blackboard.

Let $ABC$ be a triangle with circumcenter $O$, angle bisector $AD$ with $D \in BC$ and altitude $AE$ with $E \in BC$. The lines $AO$ and $BC$ meet at $I$. The circumcircle of $\triangle ADE$ meets $AB, AC$ at $F, G$ and $FG$ meets $BC$ at $H$. The circumcircles of triangles $AHI$ and $ABC$ meet at $J$. Show that $AJ$ is a symmedian in $\triangle ABC$

Let $n!_0$ denote the number obtained from $n!$ by deleting all the zeroes in the end of it decimal representation. Find all positive integers $a, b, c$, such that $a!_0+b!_0=c!_0$.