Found problems: 988
Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.
Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.
Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $
[i]Lucian Petrescu[/i]
Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]
Find all positive intger solutions of $3^x+29=2^y$.
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
Let $x, y$ be positive integers such that
$$
x^4=(x-1)\left(y^3-23\right)-1 .
$$
Find the maximum possible value of $x+y$.
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$
Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.
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.
Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$
Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.
$x^2 + 2y^2 = 1$ 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
\[a!+1=(a+1)^{(2^b)}\]
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?
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.