gcd + fibonacci
Source: IMOC 2020
September 4, 2020
number theorygreatest common divisorIMOC
Problem Statement
Let be positive integers. If satisfy that
\begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*}
is also a positive integer, show that
\begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*}
is a Fibonacci number.
Proposed by usjl