Found problems: 988
Prove that there are unique positive integers $a$ and $n$ such that \[a^{n+1}-(a+1)^{n}= 2001.\]
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$
Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of
the numbers plus one third of the other number is three times less than $1004$.
Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .
Find all pairs of positive integers $(a,b)$ such that
$$a!+b!=a^b + b^a.$$
Prove that the equation
$$\frac x{1+x^2}+\frac y{1+y^2}+\frac z{1+z^2}=\frac1{1996}$$has finitely many solutions in positive integers.
Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\]
[i]Proposed by Md. Fuad Al Alam[/i]
Find all pairs $(a,b)$ of positive integers such that $(ab)^2 - 4(a+b)$ is the square of an integer.
Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.
Let $n$ be a positive integer and $p$ a prime number $p > n + 1$.
Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$
Luu Ba Thang, Department of Mathematics, College of Education
Find all prime numbers $p$ and $q$ such that $p + q = (p -q)^3.$
Determine all pairs $(x, y)$ of integers such that \[(19a+b)^{18}+(a+b)^{18}+(19b+a)^{18}\] is a nonzero perfect square.
Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$.
[b](1)[/b] Find a $6$-digit positive integer $N$ such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
[b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$
Find the number of ordered pairs of positive integers for which
$$\frac1a+\frac1b=\frac4{2019}$$
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal:
(i) $|x|$ is a square of an integer;
(ii) $y$ is a squarefree number.
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy
$a \le b \le c$ and $abc = 2(a + b + c)$.
Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.
Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.
Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.