Found problems: 988
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
Show that the equation $x^4 +y^4 +4z^4 =1$ has infinitely many rational solutions.
Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.
Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\]
Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
Find all quadruplets $(a, b, c, d)$ of positive integers such that
\[
\left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4.
\]
Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.
Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
Determine all pairs of integers $(x,y)$ satisfying the equation
$$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$.
(N Vassiliev)
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.
Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.
For given positive integers $n$ and $p$, find neaessary and sufficient conditions for the system of equations
$$x + py = n , \\ x + y = p^2$$
to have a solution $(x, y, z)$ of positive integers. Prove also that there is at most one such solution.
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that
\[
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
\]
Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)
Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.
$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy
\begin{align*}
\vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}}
\end{align*}
$(b)$ Find all real numbers $x$ that satisfy following equation $:$
\begin{align*}
\frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6}
\end{align*}
Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.
If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$
Are there any positive integers $m$ and $n$ satisfying the equation
$m^3 = 9n^4 + 170n^2 + 289$ ?
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that
$$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$
(As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)
Given that \[34! = 95232799cd96041408476186096435ab000000_{(10)},\] determine the digits $a, b, c$, and $d$.