Difference between revisions of "2000 AIME I Problems/Problem 9"

 
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== Problem ==
 
== Problem ==
 +
The system of equations
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<center><math>\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\
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\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\
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\log_{10}(2000zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
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\end{eqnarray*}</math></center>
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 +
has two solutions <math>(x_{1},y_{1},z_{1})</math> and <math>(x_{2},y_{2},z_{2})</math>. Find <math>y_{1} + y_{2}</math>.
  
 
== Solution ==
 
== Solution ==
 +
{{solution}}
  
 
== See also ==
 
== See also ==
* [[2000 AIME I Problems]]
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{{AIME box|year=2000|n=I|num-b=8|num-a=10}}

Revision as of 19:32, 11 November 2007

Problem

The system of equations

$\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\

\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(2000zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\

\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

has two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$.

Solution

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See also

2000 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions