AIME II 2009 #10

[stupidityismygam]

Four lighthouses are located at points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] The lighthouse at [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] kilometers from the lighthouse at [unparseable or potentially dangerous latex formula] the lighthouse at [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] kilometers from the lighthouse at [unparseable or potentially dangerous latex formula] and the lighthouse at [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] kilometers from the lighthouse at [unparseable or potentially dangerous latex formula] To an observer at [unparseable or potentially dangerous latex formula] the angle determined by the lights at [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] and the angle determined by the lights at [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are equal. To an observer at [unparseable or potentially dangerous latex formula] the angle determined by the lights at [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] and the angle determined by the lights at [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are equal. The number of kilometers from [unparseable or potentially dangerous latex formula] to [unparseable or potentially dangerous latex formula] is given by [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers, and [unparseable or potentially dangerous latex formula] is not divisible by the square of any prime. Find [unparseable or potentially dangerous latex formula]

[sinkokuyou]

it's easy to think but hard to solve, it took me 2min to think and 10 min to solve it and 3 min to check the work...

the process is like this:
set the angle of BCA is [unparseable or potentially dangerous latex formula]
which then u know that they are simply asking for this:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
from the question, you know that:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
and since:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
do the same thing for cos1.5theta
which you'll get:
[unparseable or potentially dangerous latex formula]
after 5min of working, go back to the original stuff:...(too much work)
and I got:
[unparseable or potentially dangerous latex formula]
which 60+13+23=96