Mediterranean Mathematical Competition

We put $[x]=a$, $\{x\}=r$, where $0\le r<1$. Then the given inequality becomes $$\begin{gathered} \left(\frac{a+2r}{a}-\frac{a}{a+2r}\right)+\left(\frac{2a+r}{r}-\frac{r}{2a+r}\right)>\frac{9}{2} \\ \iff 2\left(\frac{r}{a}+\frac{a}{r}\right)-\left(\frac{a}{a+2r}+\frac{r}{2a+r}\right)>\frac{5}{2}. \end{gathered}$$ Since $\frac{r}{a}+\frac{a}{r}\ge 2$, it is enough to prove that $$\frac{a}{a+2r}+\frac{r}{2a+r}<\frac{3}{2}\iff 0<2a^2+11ar+2r^2\iff 2(a+r{)}^2+7ar>0,$$ which is valid.