Found problems: 3597
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers?
[i]Proposed by F. Petrov[/i]
Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that
\[ \left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2 \]
for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.
(Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)
Find all real numbers $a, b, c$ such that $x^3 + ax^2 + bx + c$ has three real roots $\alpha, \beta,\gamma$ (not necessarily all distinct) and the equation $x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3$ has roots $\alpha^3, \beta^3,\gamma^3$ .
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
Let $z_1$, $z_2$, $z_3$, $\cdots$, $z_{2021}$ be the roots of the polynomial $z^{2021}+z-1$. Evaluate $$\frac{z_1^3}{z_{1}+1}+\frac{z_2^3}{z_{2}+1}+\frac{z_3^3}{z_{3}+1}+\cdots+\frac{z_{2021}^3}{z_{2021}+1}.$$
Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)
Consider the polynomial with real coefficients:
\[ p(x) = a_{2008}x^{2008} + a_{2007}x^{2007} + \dots + a_1x + a_0 \]
and it is given that its coefficients satisfy:
\[ a_i + a_{i+1} = a_{i+2}, \quad i \in \{0,1,2,\dots,2006\} \]
If $p(1) = 2008$ and $p(-1) = 0$, compute $a_{2008} - a_0$.
Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well.
Proposed by Sutanay Bhattacharya
Given the expression
\[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\]
prove:
$(a) P_n(x)$ satisfies the identity
\[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\]
$(b) P_n(x)$ is a polynomial in $x$ of degree $n.$
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients.
Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]
Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of
\[ |(x-y)(y-z)(z-x) | \]
Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ .
(Folklore)
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$.
Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.
There are some counters in some cells of $100\times 100$ board. Call a cell [i]nice[/i] if there are an even number of counters in adjacent cells. Can exactly one cell be [i]nice[/i]?
[i]K. Knop[/i]