Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.

A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

Find all real polynomials $f$ and $g$, such that: \[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \] for all $x\in\mathbb{R}$.

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.

The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$.

Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$ [hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.

Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

Let $f(x) = x-\tfrac1{x}$, and define $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.

For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that: [list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list] Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

If $x$, $y$, $z$ are real numbers satisfying \begin{align*} (x + 1)(y + 1)(z + 1) & = 3 \\ (x + 2)(y + 2)(z + 2) & = -2 \\ (x + 3)(y + 3)(z + 3) & = -1, \end{align*} find the value of $$ (x + 20)(y + 20)(z + 20). $$

It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$; (a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$; (b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.

A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$. $ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$. $ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.

Determine the polynomial P(X) satisfying simoultaneously the conditions: a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal to the fourth power of quotient. b) P(−2) + P(2) = −34.