2012 AIME I Problems/Problem 9
Problem 9
Let
and
be positive real numbers that satisfy
The value of
can be expressed in the form
where
and
are relatively prime positive integers. Find
Solution
Since there are only two dependent equations given and three unknowns, the three expressions given can equate to any common value, so to simplify the problem let us assume with loss of generality that
Then
Solving these equations, we quickly see that
and then
Finally, our desired value is
and thus
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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