2016 AIME II Problems/Problem 15
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For let
and
. Let
be positive real numbers such that
and
. The maximum possible value of
, where
and
are relatively prime positive integers. Find
.
Solution
Note that
Substituting this into the second equation and collecting
terms, we find
Conveniently,
so we find
This is the equality case of the Cauchy-Schwarz Inequality, so
for some constant
. Summing these equations and using the facts that
and
, we find
and thus
. Hence the desired answer is
.
See Also
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
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