Found problems: 916
Find all pairs of integers $(a, b)$ that satisfy the equation
$$(a + 1)(b- 1) = a^2b^2.$$
Find all positive integers $k$ such that there exists positive integers $a, b$ such that
\[a^2+4=(k^2-4)b^2.\]
Find all the values of $n \in N$ such that $n^2 = 2^n$.
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.
Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be [i]weak[/i] if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a [i]weak[/i] integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is [i]weak[/i].
For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$
Find all integers m, n such that $m^3 = n^3 + n$.
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
Find all pairs of positive integers $(a,b)$ such that $5a^b - b = 2004$.
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation
$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
$14.$ If $3^x+2^y=985.$ and $3^x-2^y=473.$ What is the value of $xy ?$
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$
Determine all pairs $(x,y)$ of positive integers satisfying the equation \[(x+y)^{2}-2(xy)^{2}=1.\]