2010 AIME I Problems/Problem 9
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Problem
Let be the real solution of the system of equations
,
,
. The greatest possible value of
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Solution
Add the three equations to get . Now, let
.
,
and
, so
. Now cube both sides; the
terms cancel out. Solve the remaining quadratic to get
. To maximize
choose
and so the sum is
giving
.
See Also
2010 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 | ||
All AIME Problems and Solutions |
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