Found problems: 436
Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.
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 solutions of the equation $(a^a)^5 = b^b$ in positive integers.
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
Determine all pairs $(x, y)$ of positive integers satisfying
$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$
If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ?
Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.
Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$
Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.
Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.
Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$
Prove that for every $n$ there exists a solution of the equation
$$a^2 +b^2 +c^2 = 3abc$$
in natural numbers $a,b,c$ greater than $n$.
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$
Solve the following system
\[
\left\{ \begin{array}{l}
xyzu-x^3=9 \\
x+yz=\dfrac{3}{2}u \\
\end{array} \right.
\]
in positive integers $x$, $y$, $z$ and $u$.
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$
Find all pairs of integers $(m, n)$ such that $(m - n)^2 =\frac{4mn}{m + n - 1}$
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.
Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$
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Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?
Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.