Found problems: 436
Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
Solve the equation in integer numbers $$y^3-x^3=91$$
Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are
a) natural numbers,
b) integers
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.
Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation
$$p^3 + pm + 2zm = m^2 + pz + z^2$$
Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied:
$$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$
Show that there are only finitely many integral solutions to
\[
3^m - 1 = 2^n
\]
and find them.
Find all the natural numbers $x, y, z$ that satisfy simultaneously
$$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations
$$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$
Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$
then $\min\, (x,y) \ge n$.
Find all triples of integers $(x, y, z)$ such that
$$x^2 + y^2 + z^2 = 16(x + y + z).$$
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?
Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$
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 $p = q$.
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 $1$.
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers
Find all positive intger solutions of $3^x+29=2^y$.
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 $$3^x -5^y = z^2$$