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equation with rational fractional parts

Source: Romanian Nationals RMO 2005 - grade 10, problem 4

March 31, 2005
algebra proposedalgebra

Problem Statement

For α∈(0,1)\alpha \in (0,1) we consider the equation {x{x}}=α\{x\{x\}\}= \alpha. a) Prove that the equation has rational solutions if and only if there exist m,p,q∈Zm,p,q\in\mathbb{Z}, 0<p<q0<p<q, gcd⁡(p,q)=1\gcd(p,q)=1, such that α=(pq)2+mq\alpha = \left( \frac pq\right)^2 + \frac mq. b) Find a solution for α=200420052\alpha = \frac {2004}{2005^2}.
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