Found problems: 3597
Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$
Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that
\[
p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1
\]
Show that $p'(x) = nq(x)$.
Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that
$$\displaystyle P(x)=R(T(x))$$
Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$
Let $a,b,c,d$ real numbers such that $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximum value of
\[ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} \]
a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$
b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that:
$(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$
Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.
Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$.
[i]Author: Alex Zhu[/i]
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.
Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients.
We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that
$ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$.
Prove that there exists $ a,b,c\in\mathbb{C}$ such that
$ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.
Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?
Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?
Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.
Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$.
$(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$.
$(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.
Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all
integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented
in infinitely many ways of the form
$\pm P(1) \pm P(2) \pm \ldots \pm P(m)$,
for some positive integer $m$ and certain choices of $\pm$.
The product of uncommon real roots of the two polynomials $ x^4 \plus{} 2x^3 \minus{} 8x^2 \minus{} 6x \plus{} 15$ and $ x^3 \plus{} 4x^2 \minus{} x \minus{} 10$ is ?
$\textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ \minus{} 6 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None}$
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation:
$P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.