$p(x)$ 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 $a_1 = a_3$.

Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$

For a non-negative integer $n$, call a one-variable polynomial $F$ with integer coefficients $n$-[i]good [/i] if: (a) $F(0) = 1$ (b) For every positive integer $c$, $F(c) > 0$, and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$-good for any $n$.

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

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)

Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .

Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$. [i]Proposed by Gerhard Woeginger, Austria[/i]

Prove that the polynomial $P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1$, $n \in N^*$, cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to $1$.

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.

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.

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$. (Note: $1999$ is a prime number.)

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.

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.

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

It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.

Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$. For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?

Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.

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 $k$.