Found problems: 436
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.
Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.
Determine all pairs of integers $(x,y)$ satisfying the equation
$$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$.
(N Vassiliev)
For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations
$$x + py = n , \\ x + y = p^2$$
to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that
\[
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
\]
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)
Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.
Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.
If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$
Are there any positive integers $m$ and $n$ satisfying the equation
$m^3 = 9n^4 + 170n^2 + 289$ ?
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that
$$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$
(As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)
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 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 $p + q = (p -q)^3.$
Find all the solutions $(a, b, c)$ in the natural numbers, verifying $1\le a \le b \le c$, of the equation$$\frac34=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
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.
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 $(n -1)2^{n - 1} + 5 = m^2 + 4m$.
Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.