Found problems: 4275
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
Find all integer coefficient polynomials $Q$ such that
[list]
[*] $Q(n)\ge 1$ $\forall n\in \mathbb{Z}_+$.
[*] $Q(mn)$ and $Q(m)Q(n)$ have the same number of prime divisors $\forall m,n\in\mathbb{Z}_+$.
[/list]
Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
(i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$,
(ii) $ p(x)\ge0$ for all $ x$,
(iii) $ p(0)\equal{}1$
(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$.
Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that
$$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$
and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.
We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]
The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows:
$P_{1}(X)=2X$
$P_{2}(X)=2(X^2+1)$
$P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$.
Find all $n$ for which $X^2+1\mid P_{n}(X)$
Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$
Prove that: $\deg(P)$ is an even number and find $P(x).$
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$.
Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ :
$P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$
$n>1$
Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and
$(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.
Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$
holds for infinitely many real numbers $x$.
Prove that the polynomial
$$P(x) = nx^{n+2} -(n + 2)x^{n+1} + (n + 2)x-n$$
is divisible by the polynomial $(x - 1)^3$.
Let be a ring $ R $ which has the property that there exist two distinct natural numbers $ s,t $ such that for any element $ x $ of $ R, $ the equation $ x^s=x^t $ is true. Show that there exists a polynom in $ R[X] $ of degree
$$ |s-t|\left( 1+|s-t| \right) $$
such that all the elements of $ R $ are roots of it.
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.
Consider the polynomial $x^3-3x^2+10$. Let $a, b, c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.
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.
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$.
[i]Submitted by Navid Safaei, Tehran, Iran[/i]
Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find
\[\lim_{n\to\infty}f(n)^{1/n}\]
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:
\[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\]
and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\]
Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
Let $P(x)$ be a polynomial with integer coefficients such that
\[P(\sqrt{2}\sin x) = -P(\sqrt{2}\cos x)\]
for all real numbers $x$. What is the largest prime that must divide $P(2019)$?