Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that: $P(x) = b + (x-a)Q((x-a)^2)).$

Let $n$ be a positive integer. Prove that $x^n -\frac{1}{x^{n}}$ is expressible as a polynomial in $x-\frac{1}{x}$ with real coefficients if and only if $n$ is odd.

Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.

For a non-constant polynomial $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}$, we say that $P$ is symmetric if $a_{k}=a_{n-k}$ for every $k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil$. We define the weight of a non-constant polynomial $P \in \mathbb{R}[x]$, denoted by $t(P)$, as the multiplicity of its zero with the highest multiplicity. a) Prove that there exist non-constant, monic, pairwise distinct polynomials $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$, none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric. b) What is the smallest possible value of $t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)$, if $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$ are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?

The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

Let $k$ be a real number, such that the equation $kx^2 + k = 3x^2 + 2-2kx$ has two real solutions different. Determine all possible values of $k$, such that the sum of the roots of the equation is equal to the product of the roots of the equation increased by $k$.

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.

For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)

For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

Find the coefficients of $x^{17}$ and $x^{18}$ after expansion and collecting the terms of $(1+x^5+x^7)^{20}$.

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.