Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials $f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$ have integer coefficients. Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$. Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$

Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \] for all real number $x$.

Sum of all roots of the equation $$cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0$$, in interval $\left[\pi, \frac{3\pi}{2} \right]$, is equal to $21\pi$, and the sum of all roots of the equation $$sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0$$, in the same interval, is equal to $24\pi $. How many roots does the first equation have on the segment $\left[ \frac{\pi}{2}, \pi\right]$?

Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$. Prove that the polynomial \[\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}\] is irreducible in $\mathbb{Q}[X]$. [i]Beniamin Bogosel[/i]

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition: \[ p \mid uv - 1 \] we always have $p \mid f(u)f(v) - 1$.

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.

Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$. Prove that if $a_{n-2}=0$, then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$.

Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$ [i]Proposed by Michael Tang[/i]

For all $n>1$. Find all polynomials with complex coefficient and degree more than one such that $(p(x)-x)^2$ divides $p^n(x)-x$. ($p^0(x)=x , p^i(x)=p(p^{i-1}(x))$) [i]Proposed by Navid Safaie[/i]

Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$. The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$. For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others? [i]Proposed by F. Petrov, K. Sukhov[/i]

=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that \[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\} \] Are the following subsets of plane an algebraic sets? 1. A square [img]http://i36.tinypic.com/28uiaep.png[/img] 2. A closed half-circle [img]http://i37.tinypic.com/155m155.png[/img]

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.

Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$.

The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.

Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that $$x^m + x + 2 = P(P(x))$$ holds for all integers $x$.

Let $P(x)=x^{2000}-x^{1000}+1$. Do there exist distinct positive integers $a_1,\dots,a_{2001}$ such that $a_ia_j|P(a_i)P(a_j)$ for all $i\neq j$? [I]Proposed by A. Baranov[/i]

When $ x^9\minus{}x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: $ \textbf{(A)}\ \text{more than 5} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties: (i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$; (ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then $ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$; (iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$. A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties: If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$; If $ s_i\geq 0$, then $ f(S)\geq 0$; $ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$; $ f(ST)\equal{}f(S)f(T)$, $ f(S')\equal{}f(S)$ Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.