Found problems: 916
$x^2 + 2y^2 = 1$ solve in integers.
Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.
Find all the pairs of natural numbers $(a, b),$ such that
\[a!+1=(a+1)^{(2^b)}\]
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$.
(Malik Talbi)
Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.
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 $c + d$.
Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]
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 $x^n + y^n = 3^m$.
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 $x^2 +3y^2 = 1998x$.
Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$