Found problems: 4275
[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find
$ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$
For an arbitrary real y. [b](S.5)[/b]
$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$
The polynomial $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$ is written into the form $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$, where $y=x-4$, then $a_0+a_1+\cdots+a_{20}=$________.
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides
\[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\]
are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.
Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
[i]Dumitru Bușneag[/i]
Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$,
\[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$.
[i]Calvin Deng.[/i]
If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$.
[i]2015 CCA Math Bonanza Tiebreaker Round #2[/i]
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$.
[i]Proposed by Belgium[/i]
The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as:
$\textbf{(A) }2^{100}-1\qquad
\textbf{(B) }2^{100}(x-1)-(x-2)\qquad
\textbf{(C) }2^{100}(x-3)\qquad$
$\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad
\textbf{(E) }2^{100}(x+1)-(x+2)$
Let $f(x)=cx(x-1)$, where $c$ is a positive real number. We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^n(x)$ are real. What is the smallest possible value of $c$?
Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.
Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.
For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.
Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation
$$x^3 + ax^2 + bx + c = 0$$
has three real roots creating an arithmetic progression.
Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.
On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis.
[i](4 points)[/i]
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
Let $n$ be a positive integer. Suppose that $A,B,$ and $M$ are $n\times n$ matrices with real entries such that $AM=MB,$ and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.
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 $i + j + k \geq n$.
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that
\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$
In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.