Found problems: 3597
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form
\[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\]
with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?
Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying
$$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$
then each of the polynomials is zero.
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/i]
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.
If $a,b,c$ are non-zero real numbers such that \[\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},\] and \[x=\frac{(a+b)(b+c)(c+a)}{abc},\] and $x<0$, then $x$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }-2\qquad\textbf{(C) }-4\qquad\textbf{(D) }-6\qquad \textbf{(E) }-8$
Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$, such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$
For integers $0\le a\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$.
(1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$.
(2) Prove that for any positive integer $n$, $F(n)\le \frac{n-1}{3}$.
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$.
a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$.
b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.
Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$?
$\textbf{(A)}~5$
$\textbf{(B)}~\frac{25}6$
$\textbf{(C)}~29$
$\textbf{(D)}~35$
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
If a polynomial $P$ with integer coefficients has three distinct integer zeroes , then show that $P(n)\neq 1$ for any integer $n$.
(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square.
(b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$.
[i]AMM Magazine[/i]
Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.
If $ x$ is a real number such that $ x^2\minus{}x$ and $ x^n\minus{}x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation
$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
Prove that for any integer $n$, there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$.
Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions:
$\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals
$\bullet$ If $a $and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$.
Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that
$$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$