Determine all real numbers a for which there exists positive reals which satisfy the relations
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Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\displaystyle \sum_{k=1}^{5} kx_{k}=a,$ $\displaystyle \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\displaystyle \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
Source: Međunarodna matematička olimpijada, shortlist 1979
Školjka 
which satisfy the relations 
