The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

Given real numbers $b \geq a>0$, find all solutions of the system \begin{align*} &x_1^2+2ax_1+b^2=x_2,\\ &x_2^2+2ax_2+b^2=x_3,\\ &\qquad\cdots\cdots\cdots\\ &x_n^2+2ax_n+b^2=x_1. \end{align*}

We define $\mathbb F_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb F_{101}$ (the integers modulo $101$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb F_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100x+5)=100x^2+2x+15$ in $\mathbb F_{101}[x]$ because the corresponding coefficients are equal modulo $101$. We say that $f(x)\in\mathbb F_{101}[x]$ is \emph{lucky} if it has degree at most $1000$ and there exist $g(x),h(x)\in\mathbb F_{101}[x]$ such that \[f(x)=g(x)(x^{1001}-1)+h(x)^{101}-h(x)\] in $\mathbb F_{101}[x]$. Find the number of lucky polynomials. [i]Proposed by Michael Ren.[/i]

Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.

Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$. [I]Proposed by Italy[/i]

Let $f$ be a polynomial such that, for all real number $x$, $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$. Compute $f(2018)$.

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is [list=1] [*] $\frac{\pi}{2}$ [*] 0 [*] 1 [*] None of these [/list]

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.

Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]