Found problems: 988
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$.
(b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.
Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?
If $a$ and $b$ are integers and if the solutions of the system of equations
$$y - 2x - a = 0$$
$$y^2 - xy + x^2 - b = 0$$
are rational, prove that the solutions are integers.
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .
a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.
b) Show that $A$ is an infinite set.
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$
Solve the following equations in the set of natural numbers:
a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$
b) $1005^x+2011^y=1006^z$
Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.
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 pairs of positive prime numbers $(p, q)$ such that
$$p^5 + p^3 + 2 = q^2 - q.$$
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation
$$a^3 + 2017a = b^3 -2017b.$$
find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$.
Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that
\begin{align*}
x+2y+3z+7t&=a,\\
y+2z+5t&=b.
\end{align*}
Find:
a) Any quadruple of positive integers $(a, k, l, m)$ such that $a^k = a^l + a^m,$
b) Any quintuple of positive integers $(a, k, l, m, n)$ for which $a^k = a^l + a^m+a^n$
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$.
([i]Dan Schwarz[/i])
Solve in positive integers:
\[x^{y^2+1}+y^{x^2+1}=2^z\]