2005 AMC 12B Problems/Problem 23: Difference between revisions
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== Problem == | == Problem == | ||
Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which | |||
<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath> | |||
There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> | |||
<math> | |||
\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 | |||
</math> | |||
== Solution == | == Solution == | ||
Revision as of 17:12, 22 February 2010
Problem
Let 
be the set of ordered triples 
of real numbers for which
![\[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\]](http://latex.artofproblemsolving.com/c/1/1/c117b87dca578c53d5694e7f783975909325a5bc.png)
There are real numbers 
and 
such that for all ordered triples 
in we have 
What is the value of 
