2005 AMC 12B Problem 23

Let S be the set of ordered triples (x,y,z) of real numbers for which

\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1. There are real numbers a and b such that for all ordered triples (x,y,z) in S we have x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}. What is the value of a+b?

\textbf{(A)}\ \frac {15}{2} \qquad \textbf{(B)}\ \frac {29}{2} \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ \frac {39}{2} \qquad \textbf{(E)}\ 24

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.