Found problems: 3597
2023 SG Originals, Q6
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]
2003 Romania Team Selection Test, 7
Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$.
[i]Laurentiu Panaitopol[/i]
1986 Traian Lălescu, 2.1
Show that for any natural numbers $ m,n\ge 3, $ the equation $ \Delta_n (x)=0 $ has exactly two distinct solutions, where
$$ \Delta_n (x)=\begin{vmatrix}1 & 1-m & 1-m & \cdots & 1-m & 1-m & -m \\ -1 & \binom{m}{x} & 0 & \cdots & 0 & 0 & 0 \\ 0 & -1 & \binom{m}{x} & \cdots & 0 & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & -1 & \binom{m}{x} & 0 \\ 0 & 0 & 0 & \cdots & 0 & -1 & \binom{m}{x}\end{vmatrix} . $$
1989 Bundeswettbewerb Mathematik, 1
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 $n$.
1995 All-Russian Olympiad, 8
Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$.
[i]A. Galochkin, O. Ljashko[/i]
2018 Romanian Masters in Mathematics, 2
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
2011 Bosnia And Herzegovina - Regional Olympiad, 1
Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$
2011 VTRMC, Problem 7
Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.
2007 Italy TST, 3
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
1998 ITAMO, 5
Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$.
(a) Prove that there is no integer $n$ such that $P(n) = 12$.
(b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?
2017 Baltic Way, 4
A linear form in $k$ variables is an expression of the form $P(x_1,...,x_k)=a_1x_1+...+a_kx_k$ with real constants $a_1,...,a_k$. Prove that there exist a positive integer $n$ and linear forms $P_1,...,P_n$ in $2017$ variables such that the equation $$x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}$$ holds for all real numbers $x_1,...,x_{2017}$.
2018 Romanian Master of Mathematics, 2
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
PEN O Problems, 43
Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?
2020 Benelux, 1
Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.
2000 Harvard-MIT Mathematics Tournament, 9
A cubic polynomial $f$ satisfies $f(0)=0, f(1)=1, f(2)=2, f(3)=4$. What is $f(5)$?
2020 Taiwan TST Round 1, 2
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
2012 ELMO Shortlist, 7
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
1953 Miklós Schweitzer, 3
[b]3.[/b] Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$, where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$, prove that
$(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$.
[b](S. 24)[/b]
1995 USAMO, 4
Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:
(i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$
Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.
1982 AMC 12/AHSME, 1
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is
$\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$
2020 Thailand TST, 5
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
2000 Moldova Team Selection Test, 3
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
2014 Poland - Second Round, 3.
For each positive integer $n$, determine the smallest possible value of the polynomial
$$
W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx.
$$
2009 BMO TST, 4
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.
2023 Miklós Schweitzer, 8
Let $q{}$ be an arbitrary polynomial with complex coefficients which is not identically $0$ and $\Gamma_q =\{z : |q(z)| = 1\}$ be its contour line. Prove that for every point $z_0\in\Gamma_q$ there is a polynomial $p{}$ for which $|p(z_0)| = 1$ and $|p(z)|<1$ for any $z\in\Gamma_q\setminus\{z_0\}.$