Found problems: 3597
Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that
\[P(a)P(b)=P(c)\]
Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition:
\[ p \mid uv - 1 \]
we always have $p \mid f(u)f(v) - 1$.
Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.
(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$.
(b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.
Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions?
a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $
b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $
[i]Proposed by Mojtaba Zare[/i]
P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.
$f$ is a polynomial of degree $n$ with integer coefficients and $f(x)=x^2+1$ for $x=1,2,\cdot ,n$. What are the possible values for $f(0)$?
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.
Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$).
\[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]
The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$.
The real numbers $x_1,x_2,..., x_n$ and $a_0,a_1,...,a_{n-1}$ with $x_i \ne 0$ for $i \in\{1,2,.., n\}$ are such that
$$(x-x_1)(x-x_2)...(x-x_n)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$
Express $x_1^{-2}+x_2^{-2}+...+ x_n^{-2}$ in terms of $a_0,a_1,...,a_{n-1}$.
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :
$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$
Who has the winning strategy.
Proposed by [i]Alireza Haghi[/i]
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$
Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval
\[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\]
such that
\[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\]
for all polynomials $f$ with real coefficients and degree less than $n$.
The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$
Let $ f(x)$ be a polynomial with the following properties:
(i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.
Let $P$ be an odd-degree integer-coefficient polynomial. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
[i]Victor Wang[/i]
Let $a,b,c,d,e,f$ be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors $x-x_i$, with $x_i>0$ for $i=1,2,\ldots,8$. Determine all possible values of $f$.