Found problems: 3597
Evaluate the product \[(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7).\]
A polynomial $w$ of degree $n > 1$ has $n$ distinct zeros $x_1,x_2,...,x_n$. Prove that:
$$\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.$$
Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.
[i]Proposed by Hesam Rajabzadeh [/i]
Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.
Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken?
Boris Frenkin
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is
$ \textbf{(A)}\ 156 \qquad \textbf{(B)}\ 171 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 186 \qquad \textbf{(E)}\ 201$
Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.
The polynomials $f(x)$ and $g(x)$ are given. The points $A_1(f(1),g(1)),\ldots,A_n(f(n),g(n))$ are marked on the coordinate plane. It turns out that $A_1\ldots A_n$ is a regular $n{}$-gon. Prove that the degree of at least one of $f{}$ and $g{}$ is at least $n-1$.
[i]Proposed by V. Bragin[/i]
Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers:
$$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where
\[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]
For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence
\[ (r(1),r(3),\ldots,r(1023))
\]
is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.
Find the sum of the $2007$ roots of \[(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007).\]
Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?
[i]By Omid Hatami[/i]
Let $c$ be a nonzero real number. Suppose that $g(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r$ is a polynomial with integer coefficients. Suppose that the roots of $g(x)$ are $b_1,\cdots,b_r$. Let $k$ be a given positive integer. Show that there is a prime $p$ such that $p>\max(k,|c|,|c_r|)$, and moreover if $t$ is a real number between $0$ and $1$, and $j$ is one of $1,\cdots,r$, then \[|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}.\] Furthermore, if \[f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!}\] then \[\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.\]
Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i]
Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.
If $ a$, $ b$, $ c$, $ d$, and $ e$ are constants such that every $ x > 0$ satisfies
\[ \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4}
\equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2}
\plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, ,\]
then what is the value of $ a \plus{} b \plus{} c \plus{} d \plus{} e$?
A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds:
For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials.
$a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$.
$b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials.
[i]Proposed by Alireza Shavali[/i]