Found problems: 3597
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]
Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$.
a) Prove that for each $ c\in C$, $ A_c$ is infinite.
b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$.
c) Prove that each element of $ A_c$ is odd or even.
Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$.
d) $ |c| > 2$.
e) $ c\in \mathbb Q\backslash\mathbb Z$.
f) $ c$ is a non-algebraic number
g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$ and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.$$
[i](tatari/nightmare)[/i]
Determine the largest positive integer $n$ such that the following statement is true:
There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.
A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial
Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.
The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.
Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 \\
& x^3 + 5x^2-49x-245 \\
& x^3 + 39x^2 - 117x - 1435\end{align*}
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$.
Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?
Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.
Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$
has three distinct real roots which are sides of a right triangle.
Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$, $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$. For a real coefficient polynomial $f$ of degree $d$ and integer $n$, let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$. If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.
Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$
[i]G. Tusnady[/i]
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 $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$.
a)Prove that $(BA)^3=(BA)^2$.
b)Find $BA$.
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.
$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Find all polynomials $P(x)$ , such that $$P(x^3+1)=\left(P (x+1)\right)^3$$
Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.
We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that:
$i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$
$ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$
$iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$
Find all [i]interesting [/i]$n$.
[i]Proposed by Mojtaba Zare Bidaki[/i]