Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 15"
m (→Solution) |
|
(No difference)
|
Revision as of 10:26, 4 April 2012
Problem
Let denote the value of the sum
The value of can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Solution
Let
Factoring the radicand, we have
The fraction looks remarkably apt for a trigonometric substitution; namely, define
such that
. Then the RHS becomes
But
Therefore,
This gives us
So now
When we sum
, this sum now telescopes:
Therefore, the required value
giving us the desired answer of
.