# Another amazing log

Algebra Level 5

Let $$S$$ be the set of ordered triples $$(x,y,z)$$ of real numbers for which $$\log_{10} (x+y) = z$$ and $$\log_{10}(x^2+y^2) = z + 1$$. If $$a$$ and $$b$$ are real numbers such that for all ordered triples $$(x,y,z)$$ in $$S$$, we have $$x^3 + y^3 = a\cdot 10^{3z} + b\cdot10^{2z}$$. Then the value of $$(a+b)$$ is:

This is part of the set My Problems and THRILLER

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