The 3 (Not) Real Values

Algebra Level 4

{a+b+c=7a2+b2+c2=231a+b+1b+c+1a+c=31 \large \begin{cases} {a+b+c=7} \\ {a^2+b^2+c^2= 23} \\ {\frac1{a+b} +\frac1{b+c} + \frac1{a+c} = 31 } \end{cases}

If a,ba,b and cc satisfy the system of equations above, find the value of a3+b3+c3 a^3 + b^3 + c^3 .

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