This is an IMO problem from 1988, problem 6. The book does not provide a proof of this part and it is eluding me.
Let \cdots \lt s''' \lt s'' \lt s' \lt s all be nonnegative integers (a finite sequence), and let $k$ be a nonnegative integer such that \begin{align*} 0 &\leq s''\\ s'' &=\frac{(s')^2 - k}{s}\\ s'''&=\frac{(s'')^2 - k}{s'}\\ &\vdots \end{align*}
and $s''^2+s'^2$=k(s''s'+1) ,$s+s''=-s'k,ss''=s'^2-k$ Prove that $k$ is a perfect square.