Find all the possible values of a positive integer $n$ for which the expression $S_n=x^n+y^n+z^n$ is constant for all real $x,y,z$ with $xyz=1$ and $x+y+z=0$.

The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

A calculator has two operations $A$ and $B$ and initially shows the number $1$. Operation $A$ turns $x$ into $x+1$ and operation B turns $x$ into $\dfrac{x}{x+1}$. a) Show all the ways we can get the number $\dfrac{20}{23}$. b) For every rational $r \neq 1$, determine if it is possible to get $r$ using only operations $A$ and $B$.

Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

Let $ r$ be a real number such that the sequence $ (a_{n})_{n\geq 1}$ of positive real numbers satisfies the equation $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{m \plus{} 1} \leq r a_{m}$ for each positive integer $ m$. Prove that $ r \geq 4$.

Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.

$17.$ Let $a,$ $b,$ and $c.$ be such that $a+b+c=0$ and $$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}$$ is defined. What is the value of $P ?$

Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$. [i]Proposed by Giacomo Rizzo[/i]

Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f(f(xy-x))+f(x+y)=yf(x)+f(y)$$ for all real numbers $x$ and $y$.

Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and $$ \frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}. $$

For a non-constant polynomial $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}$, we say that $P$ is symmetric if $a_{k}=a_{n-k}$ for every $k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil$. We define the weight of a non-constant polynomial $P \in \mathbb{R}[x]$, denoted by $t(P)$, as the multiplicity of its zero with the highest multiplicity. a) Prove that there exist non-constant, monic, pairwise distinct polynomials $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$, none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric. b) What is the smallest possible value of $t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)$, if $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$ are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?

A polynomial $P$ in $n$ variables and real coefficients is called [i]magical[/i] if $P(\mathbb{N}^n)\subset \mathbb{N}$, and moreover the map $P: \mathbb{N}^n \to \mathbb{N}$ is a bijection. Prove that for all positive integers $n$, there are at least \[n!\cdot (C(n)-C(n-1))\] magical polynomials, where $C(n)$ is the $n$-th Catalan number. Here $\mathbb{N}=\{0,1,2,\dots\}$.

For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

Find all polynomials $f(x)$ with integer coefficients and leading coefficient equal to 1, for which $f(0)=2010$ and for each irrational $x$, $f(x)$ is also irrational.

Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}

A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds \[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \] where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let $b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$ for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.

Find all pairs $(x, y)$ of positive integers such that $x^2 + y^2 + 33^2 =2010\sqrt{x-y}$.

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

find all the function $f,g:R\rightarrow R$ such that (1)for every $x,y\in R$ we have $f(xg(y+1))+y=xf(y)+f(x+g(y))$ (2)$f(0)+g(0)=0$