Let $ P(x)$ be a polynomial with real coefficients such that $ P(x) > 0$ for all $ x \geq 0.$ Prove that there exists a positive integer n such that $ (1 \plus{} x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

Let $P$ be a real polynomial with degree $n\ge1$ such that $$P(0),P(1),P(4),P(9),\ldots,P(n^2)$$ are in $\mathbb Z$. Prove that $\forall a\in\mathbb Z,P(a^2)\in\mathbb Z$. [i]Proposed by Moubinool Omarjee[/i]

Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both \[x^{4} + 6x^{2} + 25\quad\text{and}\quad 3x^{4} + 4x^{2} + 28x + 5,\] what is $p(1)$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 8 $

A given polynomial $P(t) = t^3 + at^2 + bt + c$ has $3$ distinct real roots. If the equation $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$ has no real roots, prove that $P(2013) >\frac{1}{64}$

The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$. To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$, be determined? (b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$, be determined? (Theresia Eisenkölbl, Clemens Heuberger)

Show that there are infinitely many primes.

Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

A positive integer $n$ is given. Consider all polynomials $P(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_0$, whose coefficients are nonnegative integers, not exceeding $100$. Call $P$ [i]reducible[/i] if it can be factored into two non-constant polynomials with nonnegative integer coeffiecients, and [i]irreducible[/i] otherwise. Prove that the number of [i]irreducible[/i] polynomials is at least twice as big as the number of [i]reducible[/i] polynomials. [i]D. Zmiaikou[/i]

For a positive integer $n$, does there exist a permutation of all its positive integer divisors $(d_1 , d_2 , \ldots, d_k)$ such that the equation $d_kx^{k-1} + \ldots + d_2x + d_1 = 0$ has a rational root, if: a) $n = 2024$; b) $n = 2025$? [i]Proposed by Mykyta Kharin[/i]

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$.

Let $P(x)=x^n+a_1x^{n-1}+\ldots+a_n$ ($n\ge2$) be a polynomial with integer coefficients having $n$ real roots $b_1,\ldots,b_n$. Prove that for $x_0\ge\max\{b_1,\ldots,b_n\}$, $$P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.$$

Given a polynomial $P(x)$ with a) natural coefficients; b) integer coefficients; Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.

Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients. [i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

Let $\{F_n\}^\infty_{n=1}$ be the Fibonacci sequence and let $f$ be a polynomial of degree $1006$ such that $f(k) = F_k$ for all $k \in \{1008, \dots , 2014\}$. Prove that $$233\mid f(2015)+1.$$ [i]Note: $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n\geq 1$.[/i]

Prove that the polynomial $$P(x) = nx^{n+2} -(n + 2)x^{n+1} + (n + 2)x-n$$ is divisible by the polynomial $(x - 1)^3$.

Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]