Found problems: 3597
On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis.
[i](4 points)[/i]
Let $P(x)$ be a polynomial with real coefficients and form the polynomial
$$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$
Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation
$Q(x)=0$ has at least $2n - 1$ distinct real roots.
Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.
Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials.
[i]Laurentiu Panaitopol[/i]
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form
\[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\]
with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$
For each $n = 3^k, k > 0$ determine:
a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$
b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$
Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \]
be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$.
[i]L. Redei[/i]
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.
The polynomial $P(x) = x^3 - 6x - 2$ has three real roots, $\alpha$, $\beta$, and $\gamma$. Depending on the assignment of the roots, there exist two different quadratics $Q$ such that the graph of $y=Q(x)$ pass through the points $(\alpha,\beta)$, $(\beta,\gamma)$, and $(\gamma,\alpha)$. What is the larger of the two values of $Q(1)$?
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5
$
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
Let $ k$ be a positive integer, let $ m\equal{}2^k\plus{}1$, and let $ r\neq 1$ be a complex root of $ z^m\minus{}1\equal{}0$. Prove that there exist polynomials $ P(z)$ and $ Q(z)$ with integer coefficients such that $ (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1$.
Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$ Find all $n$ and $ a$ that satisfy the conditions.
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Discriminant of a second degree polynomial with integer coefficients cannot be
$ \textbf{(A)}\ 23
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 33
$
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]
Let $p$ be a prime and $Q(x)$ be a polynomial with integer coefficients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.
Given a polynomial $P(x)=a_{d}x^{d}+ \ldots +a_{2}x^{2}+a_{0}$ with positive integers for coefficients and degree $d\geq 2$. Consider the sequence defined by $$b_{1}=a_{0} ,b_{n+1}=P(b_{n}) $$ for $n \geq 1$ . Prove that for all $n \geq 2$ there exists a prime $p$ such that $p$ divides $b_{n}$ but does not divide $b_{1}b_{2} \ldots b_{n-1}$.
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.