Found problems: 3597
Find all polynomials $P(x)$ with real coefficients such that for all real $x$ holds the equality $$(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .$$
Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.
Let $f(x) = x^4 + 2x^3 - x - 1$.
(a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.
(b) Find the exact values of the 4 roots of $f(x)$.
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$
[i] Ivan Landgev[/i]
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$ \textbf{(A)}\ x^2-bx-ac=0$
$\qquad\textbf{(B)}\ x^2-bx+ac=0$
$\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$
${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$
${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $
Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?
Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds:
$ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]
$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.
Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$
(A)$5$
(B)$6$
(C)$7$
(D)$8$
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.
Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$
a) is finite
b) does not exceed $n$.
Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$.
I.Voronovich
A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.
A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.
Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.
Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].