Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. Determine all polynomials $Q(x) \in \mathbb{Z}[x]$, such that for every positive integer $n$, there exists a polynomial $R_n(x) \in \mathbb{Z}[x]$ satisfies $$Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).$$

Let $\alpha$ and $\beta$ be positive rational numbers so that $\alpha+\beta\sqrt{5}$ is a root of some polynomial $x^2+ax+b$ where $a$ and $b$ are integers. What is the smallest possible value of $\alpha\beta$?

Let $a, b$ be given two real number with $a \ne 0$. Find all polynomials $P$ with real coefficients such that $x P(x - a) = (x - b)P(x)$ for all $x\in R$

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$? $\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

Find all positive integers $n$ such that there exists a monic polynomial $P(x)$ of degree $n$ with integers coefficients satisfying $$P(a)P(b)\neq P(c)$$ for all integers $a,b,c$.

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

Prove: there are polynomials $S_1, S_2, \ldots$ in the variables $x_1, x_2, \ldots,y_1, y_2,\ldots$ with integer coefficients satisfying, for every integer $n \ge 1$, $$\sum_{d \mid n} d \cdot S_d ^{n/d}=\sum_{d \mid n} d \cdot (x_d ^{n/d}+y_d ^{n/d}) \quad (*)$$ Here, the sums run through the positive divisors $d$ of $n$. For example, the first two polynomials are $S_1 = x_1 + y_1$ and $S_2 = x_2 + y_2 - x_1y_1$, which verify identity $(*)$ for $n = 2$: $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \cdot(x_2 + y_2)$.

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.

Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$. Prove that the polynomial \[\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}\] is irreducible in $\mathbb{Q}[X]$. [i]Beniamin Bogosel[/i]

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

Is there a real number $\alpha$ such that $\cos\alpha$ is irrational but $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are all rational? (Author: V. Senderov)

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.