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Adding the given equations, we get $$a^3+b^3+c^3=2(a+b+c)+15.$$We see that $a,$ $b,$ and $c$ are the roots of $x^3-2x-5=0.$ By Vieta's formulas, $a+b+c=0,$ so $a^3+b^3+c^3=\boxed{15}.$