AIME #7 1988

[stupidityismygam]

pretty easy...this is based on a request by roel...ill post a problem and once it is solved i will post another one....ill start with easy ones like 1980's 6-10, and then move on to 2000-2007 6-10, 1990-1999 6-10, and then eventually moving up to 11-15...in the beginning i will keep away from diagram needed geometry...though trig based geometry (like this problem) will be used.

For those of you not familiar with the AMC/AIME series, the AIME is a 3 hour, 15 problem test that is qualified for by taking the AMC tests earlier in the year. All answers on the AIME are integers between 0 and 999 inclusive. Have fun!!

So here is the problem

In triangle [unparseable or potentially dangerous latex formula] [unparseable or potentially dangerous latex formula] and the altitude from [unparseable or potentially dangerous latex formula] divides [unparseable or potentially dangerous latex formula] into segments of length [unparseable or potentially dangerous latex formula]and [unparseable or potentially dangerous latex formula] What is the area of triangle [unparseable or potentially dangerous latex formula]

[Roel]

It might be hard to explain without a picture but here it goes...

Lets call the altitude [unparseable or potentially dangerous latex formula] and the intersection [unparseable or potentially dangerous latex formula]. There are 2 right triangles formed, [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. We can calculate that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula], for easier notation let [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula].

Notice that [unparseable or potentially dangerous latex formula] so we have

[unparseable or potentially dangerous latex formula] substituting gives
[unparseable or potentially dangerous latex formula].

After cross-multiplying, multiplying both sides by [unparseable or potentially dangerous latex formula], and sending everything to one side we get...
[unparseable or potentially dangerous latex formula] which reduces...
[unparseable or potentially dangerous latex formula]
Factoring:
[unparseable or potentially dangerous latex formula]
Since our [unparseable or potentially dangerous latex formula](altitude) needs to be positive we say its [unparseable or potentially dangerous latex formula].
Now its just [unparseable or potentially dangerous latex formula]

[stupidityismygam]

correct, ill post the next problem soon