I have recently watched this Numberphile video explaining how Zvezdelina Stankova solved the notorious 1988 IMO problem 6 as a student.
(Most of you will know this problem)
Let $x$ and $y$ be positive integers, if $k$ is an integer, prove that $k$ is a perfect square.
$$ \frac{x^2+y^2}{xy+1} = k $$
In this video, she says that her solution included a "cubic polynomial" and "induction", but she can't remember her solution.
She added, "I remember my teammate solved this problem with quadratic polynomial and induction, and I remember his solution to this day."
His solution, obviously, is the well-known Vieta's jumping method, and it doesn't involve cubic polynomials.
How can this problem involve cubic polynomial while solving?
Questions on this problem are maybe outdated, but I still want to find answers or some hints.