Found problems: 3597
Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.
Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]
Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by \[x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) \quad n\geq 1\]
for every positive integer $m$ is a divisor of some non-zero element of $(x_n )$
At least one of the coefficients of a polynomial $P(x)$ is negative. Can all of the coefficients of all of its powers $(P(x))^n$, $n > 1$, be positive?
(0 Kryzhanovskij)
Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$
[i]Proposed by N. Safaei (Iran)[/i]
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.
Let $p(x)=x^4-4x^3+2x^2+ax+b$. Suppose that for every root $\lambda$ of $p$, $\frac{1}{\lambda}$ is also a root of $p$. Then $a+b=$
[list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list]
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.
Let $f(x) = x-\tfrac1{x}$, and defi ne $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.
Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$.
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i]
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.
The polynomial $x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k$ has $k$ distinct real roots. Show that $a_1^2 > \frac{2ka_2}{k-1}$.
$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that
$$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$
$\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite.
$\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$.
[i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b].
[color=#3D9186]#1735[/color]
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 $0=a_{k}=a_{k+1}= ...$ :
$P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$
$n>1$
Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients.
Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$
For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds:
For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which
\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]
a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$.
b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.
Given quadratic trinomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$, where $a>c$. It is known that for every real $t$ and $s$ with $t+s=1$ the polynomial $B(x)=tP(x)+sQ(x)$ has at least one real root.
Prove that $bc \geq ad$.
Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016.
$P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.