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Difference between revisions of "2005 AMC 12B Problems/Problem 23"
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== Problem ==
 
== Problem ==
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Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which
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<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath>
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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>
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<math>
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\textbf{(A)}\ \frac {15}{2} \qquad
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\textbf{(B)}\ \frac {29}{2} \qquad
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\textbf{(C)}\ 15 \qquad
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\textbf{(D)}\ \frac {39}{2} \qquad
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\textbf{(E)}\ 24
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</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 17:12, 22 February 2010

Problem

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 $a+b?$

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

Solution

See also

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