2000 AIME I Problems/Problem 9
Problem
The system of equations
\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)has two solutions and . Find .
Solution
Since , we can reduce the equations to a more recognizable form:
Let be respectively. Using SFFT, the above equations become (*)
From here, multiplying the three equations gives
Dividing the third equation of (*) from this equation, . This gives , and the answer is .
Alternatively, at (*), notice that the RHS of the first two equations are the same, so . Substituting this into the third equation gives , which if we solve backwards for will give us the same answer.
See also
2000 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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