Find all integers $n \ge 3$ for which the polynomial \[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\] can be written as a product of two non-constant polynomials with integer coefficients.

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

If $a, b, c, d$ are the solutions of the equation $x^4-kx-15=0$, find the equation whose solutions are $\frac{a+b+c}{d^2}, \frac{a+b+d}{c^2}, \frac{a+c+d}{b^2}, \frac{b+c+d}{a^2}$.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?

For integers $0\le a\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$. (1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$. (2) Prove that for any positive integer $n$, $F(n)\le \frac{n-1}{3}$.

Given a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + ...+ a_1x + a_0$ of real coefficients. Suppose that $P(x)$ has $n$ real roots (not necessarily distinct), and there exists a positive integer $k$ such that $a_k = a_{k-1} = 0$. Prove that $P(x)$ has a real root of multiplicity $k + 1$.

Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that \[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.

For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\ Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\ Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. [i]Proposed by Morteza Saghafian[/i]

A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

Let the polynomial $P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ has at least one real root and $a_0 \ne 0$. Prove that, consequently crossing out the monomials in the notation $P(x)$ in some order, we can obtain the number $a_0$ from it so that each intermediate polynomial also has at least one real root.

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.