Found problems: 436
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,
for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$.
(Malik Talbi)
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.
Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.
Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.
Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.