Numbers and Reciprocals

Algebra Level 4

Three positive real numbers p,q,rp,q,r satisfy

1p+q=21q+r=31r+p=5 \begin{aligned} \frac { 1 }{ p } +q &=& 2\\ \frac { 1 }{ q } +r &=& 3\\ \frac { 1 }{ r } +p &=& 5\\ \end{aligned}

Find the sum of all possible values of pp.

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