There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.

Does there exist an irreducible two variable polynomial $f(x,y)\in \mathbb{Q}[x,y]$ such that it has only four roots $(0,1),(1,0),(0,-1),(-1,0)$ on the unit circle.

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

Prove that when $ f, m, n $, are any non-negative integers, then the polynomial $$ P(x) = x^{3k+2} + x^{3m+1} + x^{3n}$$ is divisible by the polynomial $ x^2 + x + 1 $.

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by \[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\] for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$

Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is: $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $

We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$. Let $P(x)$ be a good polynomial of degree $d$, and let $N_d$ be the product of all composite numbers not exceeding $d$. Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.

Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$. $\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

The third-degree polynomial $P(x)=x^3+2x^2-3x-5$ has the three roots $a$, $b$ and $c$. State a third degree polynomial with roots $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$.

Let $P(X)=aX^3-\frac16 X$ where $a\in\mathbb{R}$. [b]1)[/b] Determine $a$ such that for every $\alpha\in\mathbb{Z}$ we have $P(\alpha)\in\mathbb{Z}$. [b]2)[/b] Show that if $a$ is irrational then for every $0<u<v<1$ there exists $n\in\mathbb{Z}$ such that \[u<P(n)-\lfloor P(n)\rfloor <v.\] Generalize the problem!

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

Let $f(x)$ and $g(x)$ be non-constant polynomials with integer positive coefficients, $m$ and $n$ are given natural numbers. Prove that there exists infinitely many natural numbers $k$ for which the numbers $$f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)$$ are composite. [i]I. Tonov[/i]

Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$. Prove that $0<a<1<b<3<c<4$ [hide="Solution"] Let $abc=k$, then $a,b,c\ (a<b<c)$ are the roots of cubic equation $x^3-6x^2+9x-k=0\Longleftrightarrow x(x-3)^2=k$ that is to say, $a,b,c\ (a<b<c)$ are the $x$-coordinates of the interception of points between $y=x(x-3)^2$ and $y=k$. $y=x(x-3)^2$ have local maximuml value of $4$ at $x=1$ and local minimum value of $0$ at $x=3$. Since the $x$-coordinate of the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at local maximum point $(1,4)$ is a point $(4,4)$,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and $y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are $0<a<1,1<b<3,3<c<4 $,we are done.[/hide]

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$, [b](a)[/b] $|f(x)| \leq 5/4$, [b](b)[/b] $|g(x)| \leq 2$.

Determine the polynomial $$f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 $$ of smallest degree such that $a_i \in \{-1,0,1\}$ for $0\leq i \leq k-1$ and $f(n)$ is divisible by $30$ for all positive integers $n$.

Let $f$ be a polynomial with degree at most $n-1$. Show that $$ \sum_{k=0}^n\left(\begin{array}{l} n \\ k \end{array}\right)(-1)^k f(k)=0 $$

What is the maximal possible number of roots on the interval (0,1) for a polynomial of degree 2022 with integer coefficients and with the leading coefficient equal to 1?

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.

Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?