Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.

Let $P(x)$ be a polynomial with degree $2017$ such that $P(k) =\frac{k}{k + 1}$, $\forall k = 0, 1, 2, ..., 2017$ . Calculate $P(2018)$.

Let $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$. If $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$ for distinct numbers $a,b,c$, what are all possible values of $a+b+c$? [i]2019 CCA Math Bonanza Individual Round #12[/i]

Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.

Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. Find $P(x^2 + 1)$.

Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Find all polynomials $p(x)$ such that for all reals $a, b$ and $c$, with $a+b+c=0$, satisfies $$p(a^3)+p(b^3)+p(c^3)=3p(abc)$$

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.

Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$

Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root. Find all possible values of $P(0)$

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

Find all polynomials $P_n(x)$ of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1),$$with integer coefficients, such that its roots $x_1,x_2,\ldots,x_n$ satisfy $k\le x_k\le k+1$ for $k=1,2,\ldots,n$.

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

Polynomial $ P(t)$ is such that for all real $ x$, \[ P(\sin x) \plus{} P(\cos x) \equal{} 1. \] What can be the degree of this polynomial?

Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$: $xP(x-c) = (x - 2014)P(x)$

What is the sum of the reciprocals of the roots of the equation \[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? \] $ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after $2014$ tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of $x^{2013}$ is $2014$, and that the constant term is $1$. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)

The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$. (a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$. (b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.

Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different. For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$. Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.

Let $f(x)$ and $g(x)$ be two polynomials with real coefficients such that for infinitely many rational values of $x$, the fraction $\frac{f(x)}{g(x)}$ is rational. Prove that $\frac{f(x)}{g(x)}$ can be written as the ratio of two polynomials with rational coefficients.