Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

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]

Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$ Proposed by [i]Shayan Talaei[/i]

Let $S = \{f(a, b) | a, b = 1,2,3, 4$ and $a \ne b\}$, and consider all nonzero polynomials $p(X,Y )$ with integer coefficients such that $p(a, b) = 0$ for every element $(a,b)$ in $S$. (a) What is the minimal degree of such polynomial $p(X, Y )$ ? (b) Determine all such polynomials $p(X, Y )$ with minimal degree.

Let $P$ be an odd-degree integer-coefficient polynomial. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root. [i]Victor Wang[/i]

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 $(P(x),a)$.

Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.

Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying $$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$

Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.

$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.

Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.

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 $m,n$ be positive integers and $p$ be a prime number. Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.

Prove that there exists a unique polynomial function f with positive integer coefficients such that $f(1) = 6$ and $f(2) = 2016$.

Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.

Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and that $f(0)$ is a square-free number. Prove that the polynomial $g(x) = f(x^3)$ is also irreducible

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$. (a) Prove that there is no integer $n$ such that $P(n) = 12$. (b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.

Find a polynomial $P$ of lowest possible degree such that (a) $P$ has integer coefficients, (b) all roots of $P$ are integers, (c) $P(0) = -1$, (d) $P(3) = 128$.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.