Let $f(X)$ be a monic irreducible polynomial over $\mathbb{Z}$; therefore, by Gauss's Lemma, $f$ is also irreducible over $\mathbb{Q}$ (you may assume this). Moreover, assume $f(X) \mid f\left(X^2+n\right)$ where $n$ is an integer such that $n \notin\{-1,0,1\}$. Show that $n^2 \nmid f(0)$.

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.

Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds: $ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$ $ \textbf{(B)}\ \text{for all values of }x$ $ \textbf{(C)}\ \text{only when }x \equal{} 0$ $ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$ $ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$

The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and [*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.

Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]

For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid $$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$ and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$? (Serbia)

Two players alternately replace the stars in the expression \[*x^{2000}+*x^{1999}+...+*x+1 \] by real numbers. The player who makes the last move loses if the resulting polynomial has a real root $t$ with $|t| < 1$, and wins otherwise. Give a winning strategy for one of the players.

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

The following figure shows a [i]walk[/i] of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("$O$", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties: [list] [*] It starts at the origin, labelled $O$. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.[/list] Let's call a walk with these three properties a [i]northern walk[/i]. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?