Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2,...$ and $b_0, b_1, b_2, ...$ of real numbers by $a_{n+1}= a_n + b_n$ and $b_{n+1}=a_nb_n$ for all $n = 0, 1, 2,...$. Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022}= a_0$ and $b_{2022}= b_0$.

Let $$f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2$$ (here there are $n$ brackets $( )$). Find $f''(0)$

Does there exist a positive integer $n{}$ such that for any real $x{}$ and $y{}$ there exist real numbers $a_1, \ldots , a_n$ satisfying \[x=a_1+\cdots+a_n\text{ and }y=\frac{1}{a_1}+\cdots+\frac{1}{a_n}?\] [i]Artemiy Sokolov[/i]

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied \[f(x+y) + f(x)f(y) = f(xy) + 1 \] $\forall x, y \in \mathbb{R}$

Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).

Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$: \[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\] a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points) \[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\] b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points) \[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\] c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that: \[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\] Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)

Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

Evaluate $1001^3 - 1000^3$

Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$. [i]Proposed by Evan Chen[/i]

Non-zero real numbers $a$, $b$ and $c$ are such that any two of the three equations $ax^{11}+bx^4+c=0$, $bx^{11}+cx^4+a=0$, $cx^{11}+ax^4+b=0$ have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)

Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties: (a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$, (b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite

Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$. Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .

We say that an integer $m$ is a perfect power if there are $a\in\mathbf{Z}$, $b\in\mathbf{N}$ with $b > 1$ such that $m = a^b$. Find all polynomials $P\in\mathbf{Z}[x]$ such that $P(n)$ is a perfect power for every $n\in\mathbf{N}$.

Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that $$(a - b)(b - c)(a- c) \le 2.$$ When does equality hold? [i](Karl Czakler)[/i]

Find all pairs of positive integers \( (a, b) \) such that \( f(x) = x \) is the only function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, \( f^n(x) \) represents the function obtained by applying \( f \) \( n \) times to \( x \). That is, \( f^1(x) = f(x) \) and \( f^{n+1}(x) = f(f^n(x))\) for all \(n \geq 1\).

The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \(A\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \(AJABCHCHIJA\), which has \(10\) steps. Let \(n\) be the number of paths with \(15\) steps that begin and end at point \(A\). Find the remainder when \(n\) is divided by \(1000\). [asy] unitsize(32); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } real s = 4; dot(1 * dir( 90), linewidth(s)); dot(1 * dir(162), linewidth(s)); dot(1 * dir(234), linewidth(s)); dot(1 * dir(306), linewidth(s)); dot(1 * dir(378), linewidth(s)); dot(2 * dir(378), linewidth(s)); dot(2 * dir(306), linewidth(s)); dot(2 * dir(234), linewidth(s)); dot(2 * dir(162), linewidth(s)); dot(2 * dir( 90), linewidth(s)); defaultpen(fontsize(10pt)); real r = 0.05; label("$A$", (1-r) * dir( 90), -dir( 90)); label("$B$", (1-r) * dir(162), -dir(162)); label("$C$", (1-r) * dir(234), -dir(234)); label("$D$", (1-r) * dir(306), -dir(306)); label("$E$", (1-r) * dir(378), -dir(378)); label("$F$", (2+r) * dir(378), dir(378)); label("$G$", (2+r) * dir(306), dir(306)); label("$H$", (2+r) * dir(234), dir(234)); label("$I$", (2+r) * dir(162), dir(162)); label("$J$", (2+r) * dir( 90), dir( 90)); [/asy]

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?

Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0$ are of module $ 1$, then so are the roots of the equation $ x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0$.

(a) Prove that for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$, \[ \max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0. \] (b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$.

Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$. I.Voronovich