Found problems: 4275
Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.
Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.
Let $n \ge 3$ be an integer. Let $\mathcal{P}$ denote the set of vertices of a regular $n$-gon on the plane. A polynomial $f(x, y)$ of two variables with real coefficients is called $\textit{regular}$ if $$\mathcal{P} = \{(u, v) \in \mathbb{R}^2 \, | \, f(u, v) = 0 \}.$$ Find the smallest possible value of the degree of a regular polynomial.
[i]Proposed by Navid Safaei[/i]
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is
$\textbf{(A)} \ 2 \qquad \textbf{(B)} \ -2 \qquad \textbf{(C)} \ -2x-2 \qquad \textbf{(D)} \ 2x+2 \qquad \textbf{(E)} \ 2x-2$
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.
Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients.
Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: \[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \]
Let \(m,n\) and \(a\) be positive integers. Lumis has \(m\) cards, each with the number \(n\) written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than \(\displaystyle\frac{n}{2^m}\). Prove that if \(n\) is replaced everywhere by \(a\), then the resulting expression will have the same value as before or will be undefined due to division by zero.
[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form
$\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).
From the coefficients $a_{kn}$ compose the sequence of polynomials
$P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$)
and find a closed form for the function
$F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$
[b](S. 22)[/b]
Consider integers $m\ge 2$ and $n\ge 1$.
Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .
Let $\mathcal K$ be a field with $2^{n}$ elements, $n \in \mathbb N^\ast$, and $f$ be the polynomial $X^{4}+X+1$. Prove that:
(a) if $n$ is even, then $f$ is reducible in $\mathcal K[X]$;
(b) if $n$ is odd, then $f$ is irreducible in $\mathcal K[X]$.
[hide="Remark."]I saw the official solution and it wasn't that difficult, but I just couldn't solve this bloody problem.[/hide]
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.
Let $a,b,c,d$ be the four roots of the polynomial
\[x^4+3x^3-x^2+x-2.\]
Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of
\[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]
$P(x)$ is a monoic polynomial with integer coefficients so that there exists monoic integer coefficients polynomials $p_1(x),p_2(x),\dots ,p_n(x)$ so that for any natural number $x$ there exist an index $j$ and a natural number $y$ so that $p_j(y)=P(x)$ and also $deg(p_j) \ge deg(P)$ for all $j$.Show that there exist an index $i$ and an integer $k$ so that $P(x)=p_i(x+k)$.
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$.
Find $P(x^2 + 1)$.
Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that:
\[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]
Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .
Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.
Given that $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.
There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.