Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$

Let $p$ be a prime number. Let $\mathbb F_p$ denote the integers modulo $p$, and let $\mathbb F_p[x]$ be the set of polynomials with coefficients in $\mathbb F_p$. Define $\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$ by \[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \] Prove that for nonzero polynomials $F,G \in \mathbb F_p[x]$, \[ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). \] Here, a polynomial $Q$ divides $P$ if there exists $R \in \mathbb F_p[x]$ such that $P(x) - Q(x) R(x)$ is the polynomial with all coefficients $0$ (with all addition and multiplication in the coefficients taken modulo $p$), and the gcd of two polynomials is the highest degree polynomial with leading coefficient $1$ which divides both of them. A non-zero polynomial is a polynomial with not all coefficients $0$. As an example of multiplication, $(x+1)(x+2)(x+3) = x^3+x^2+x+1$ in $\mathbb F_5[x]$. [i]Proposed by Mark Sellke[/i]

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.

Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that $$x^m + x + 2 = P(P(x))$$ holds for all integers $x$.

Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]