RO-NMO 2005 Problem 104

For $\alpha \in (0,1)$ we consider the equation $\{x\{x\}\}= \alpha$ .

a) Prove that the equation has rational solutions if and only if there exist $m,p,q\in\mathbb{Z}$ , $0<p<q$ , $\gcd(p,q)=1$ , such that $\alpha = \left( \frac pq\right)^2 + \frac mq$ . b) Find a solution for $\alpha = \frac {2004}{2005^2}$ .