Let \[p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0\] and \[q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0\] be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$: \[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]

Assume natural number $n\ge2$. Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses index $i$ from the set $\{0,1,\cdots,n\}$, such that none of the two players had chosen this index in the previous turns; also this player in this turn chooses nonzero rational number $a_i$ too. Ali performs the first turn. The game ends when all the indices $i\in\{0,1,\cdots,n\}$ were chosen. In the end, from the chosen numbers the following polynomial is built: $$P(x)=a_nx^n+\cdots+a_1x+a_0$$ Ali's goal is that the preceding polynomial has a rational root and Amin's goal is that to prevent this matter. Find all $n\ge2$ such that Ali can play in a way to be sure independent of how Amin plays achieves his goal.

Suppose $p(x) \in \mathbb{Z}[x]$ and $P(a)P(b)=-(a-b)^2$ for some distinct $a, b \in \mathbb{Z}$. Prove that $P(a)+P(b)=0$.

Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property: [list] [*]$f(0)$ and $f(1)$ are both odd. [*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$ [/list] Prove that there are infinitely many prime numbers dividing at least one element of the sequence. [i]Proposed by Sayandeep Shee[/i]

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

Let $a$ and $b$ be natural numbers. Prove that the number of polynomials $P(x)$ with integer coefficients such that $|P(n)| \leq a^n$ for every natural number $n \geq b$ is finite.

The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of $a$, $b$, and $c$): $ \textbf{(A) }\text{irreducible}\qquad\textbf{(B) }\text{reducible to negative 1}\qquad$ $\textbf{(C) }\text{reducible to a polynomial of three terms} \qquad\textbf{(D) }\text{reducible to} \frac{a-b+c}{a+b-c} \qquad\textbf{(E) }\text{reducible to} \frac{a+b-c}{a-b+c} $

Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$. a) Find all values of $P$ lying between 1 and 100. b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$? $\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$

Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$

Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$. Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.

For numbers $a,b \in \mathbb{R}$ we consider the sets: $$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$ Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$: $$ P(r) \in A \iff Q(r) \in B$$

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .

Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.

Find a necessary and sufficient condition on the natural number $ n$ for the equation \[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0 \] to have a integral root.

Determine all polynomials $P(x)$ with real coefficients such that \[(x+1)P(x-1)-(x-1)P(x)\] is a constant polynomial.

Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation: $ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad \textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad \textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\ \textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad \textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$

Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$. (a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$. (b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

Find all polynomials $P\in \mathbb C[X]$ such that \[P(X^{2})=P(X)^{2}+2P(X)\]