$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$

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 $m$ .

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.

Let $k$ be a real number. Define on the set of reals the operation $x*y$= $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$. Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$.suppose that $[(x_1*x_2)*x_3]*x_4=1$. Find all possible values of $k$

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

Let $ a$, $ b$, $ c$ be integers each with absolute value less than or equal to $ 10$. The cubic polynomial $ f(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ satisfies the property \[ \Big|f\left(2 \plus{} \sqrt 3\right)\Big| < 0.0001. \] Determine if $ 2 \plus{} \sqrt 3$ is a root of $ f$.

For each non-constant integer polynomial $P(x)$, let’s define $$M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.$$ 1. Find the minimum value of $M_{P(x)}$ when deg $P(x) = 1$. 2. Suppose that $P(x) \in Z[x]$ when deg $P(x) = n$ and $2 \le n \le 2022$. Prove that $M_{P(x)} \ge 1011$.

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$? $\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions: [b](i)[/b] if $f \in A$ and $\deg( f ) \leq 1$, then $f(x) = x - 1$; [b](ii)[/b] if $f \in A$ and $\deg( f ) \geq 2$, then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$; [b](iii)[/b] for every $g, h \in A$, both $x^{2+\deg(g)} + xg(x) -1$ and $x^{1+\deg(g)}g(x) + h(x)$ belong to $A.$ Let $R_n(f)$ be the remainder of the Euclidean division of the polynomial $f(x)$ by $x^n$. Prove that for all $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$, and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$.

A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.

Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation \[f(x)f(y)-af(xy)=x+y\] , for every real numbers $x,y$. Find all possible real values of $a$.

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\textbf{(A) }3\qquad \textbf{(B) }4\qquad \textbf{(C) }5\qquad \textbf{(D) }6\qquad \textbf{(E) }11$

Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.

Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$

Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies \[x \in \mathbb R \iff p(x) \in \mathbb R.\] Show that $n=1$

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$ a) is finite b) does not exceed $n$.

Are there positive real numbers $x$, $y$ and $z$ such that $ x^4 + y^4 + z^4 = 13\text{,} $ $ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $ $ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $ [i]Proposed by Matko Ljulj.[/i]

$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.