Does there exist a function $f:\mathbb N\to\mathbb N$ such that $$ f(f(n+1))=f(f(n))+2^{n-1} $$for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.) (I. Gorodnin)

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ (Folklore)

Given the equation $$ a^b\cdot b^c=c^a $$in positive integers $a$, $b$, and $c$. (i) Prove that any prime divisor of $a$ divides $b$ as well. (ii) Solve the equation under the assumption $b\ge a$. (iii) Prove that the equation has infinitely many solutions. (I. Voronovich)

Let the sequence $(a_n)$ be constructed in the following way: $$ a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots. $$Prove that $a_{180}>19$. (Folklore)

Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$. Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that $$ p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1. $$ (I. Gorodnin)

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. (M. Berindeanu, RMC 2018 book)

$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones. (O. Izhboldin)

Cells of $11\times 11$ table are colored with $n$ colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than $7$ and not greater than $13$. Prove that there exists at least one row or column which contains cells of at least four different colors. (N. Sedrakyan)

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. Proposed by Mongolia

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called good, if the following conditions hold: each triangle from $T$ is inscribed in $\omega$; no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \]Is it possible that both $xy$ and $zt$ are perfect squares?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$. Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?) (Folklore)

Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively. Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$. (M. Karpuk)

A polygon (not necessarily convex) on the coordinate plane is called plump if it satisfies the following conditions: $\bullet$ coordinates of vertices are integers; $\bullet$ each side forms an angle of $0^\circ$, $90^\circ$, or $45^\circ$ with the abscissa axis; $\bullet$ internal angles belong to the interval $[90^\circ, 270^\circ]$. Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons. (A. Yuran)

Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$. Prove that the lines $BK$ and $AC$ are perpendicular. (M. Karpuk)

The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain. Find the value of $\frac{A}{B}$. (I. Voronovich)

Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]forms an arithmetic progression. Prove that the terms of the sequence are equal.

The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$. Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$. (A. Voidelevich)

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that $$ (a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n. $$ (B. Serankou, M. Karpuk)

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions: 1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$; 2. $f$ takes all integer values. (I. Voronovich)

Prove that for $n>1$ , $n$ does not divide $2^{n-1}+1$