Found problems: 4275
Find all pairs of polynomials $P(x),Q(x)$ with integer coefficients such that $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$ for all real numbers $x$
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$
Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$
The $100$th degree polynomial $P(x)$ satisfies $P(2^k) = k$ for $k = 0, 1, . . . 100$. Let $a$ denote the leading coefficient of $P(x)$. Find the unique integer $M$ such that $2^M < |a| < 2^{M+1}$.
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Solve this equation
$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$
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.
Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$
with rational entries and size $ 2019 $ such that:
a) $ A ^ 3 + 6A ^ 2-2I = 0 $?
b) $ A ^ 4 + 6A ^ 3-2I = 0 $?
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.
Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$.
Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.
Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions:
$$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$
Find all polynomials
$f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$
with the following properties
(a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and
(b) all the roots of the equation $f(x)=0$ are real.
Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.
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 $3$.
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac1 2$ , then compute $a^2 + b^2$.
For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$?
$\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$
Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.
Find the sum of the coefficients of the polynomial $(63x-61)^4$.
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
[i]M. Didin[/i]
Let $n\geq 2$ be a positive integer and
\begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*}
be a polynomial with integer coefficients, such that $\mid c_n \mid$ is a prime number and
\begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*}
Prove that the polynomial $P(X)$ is irreducible in the $\mathbb{Z}[x]$
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f (0) = 2004$.
2. If $x$ is irrational, then $f (x)$ is also irrational.
(Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)
If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that
$$a_1f(1)+\ldots+a_tf(t)=k.$$
Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$.
[i]Alternative formulation[/i]:
Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial
\[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}.
\]
Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$
[i]Proposed by Federico Ardila, Colombia[/i]
Find all complex roots (with multiplicities) of the polynomial
$$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$