2016 AIME II Problems/Problem 3
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Problem
Let and
be real numbers satisfying the system
\begin{align*}
\log_2(xyz-3+\log_5 x)&=5,\\
\log_3(xyz-3+\log_5 y)&=4,\\
\log_4(xyz-3+\log_5 z)&=4.\\
\end{align*}
Find the value of
.
Solution
First, we get rid of logs by taking powers: ,
, and
. Adding all the equations up and using the
property, we have
, so we have
. Solving for
by substituting
for
in each equation, we get
, so adding all the absolute values we have
.
Note: because we know
has to be a power of
, and so it is not hard to test values in the equation
in order to achieve desired value for
.
See also
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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