AIME #9 1991

[stupidityismygam]

Suppose that [unparseable or potentially dangerous latex formula] and that [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] is in lowest terms. Find [unparseable or potentially dangerous latex formula]

I think I am going to stop posting these...if I don't get some interest...

Even if you don't have a complete solution post your work...so that i know that there is some interest...otherwise, I am just wasting my time finding problems and posting solutions..

[Quelloquialism]

If you have the time, don't stop posting them...my problem lately has been a lack of free time combined with a lack of ability to solve (like this one, which I've spent 45 minutes on, but I haven't figured anything out at all).

[Quelloquialism]

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
:: [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
:: [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] so [unparseable or potentially dangerous latex formula]

Hehe...what an awful solution. I figured out what I was doing wrong before...I got my Pythagorean identity wrong, ha. I'm dumb.

[stupidityismygam]

yay good job...

It's not that I don't have the time...it's just it didn't seem like there was any interest, but there obviously is. I'll continue to post and then like a week later post a solution if it is solutionless.

I didn't think of the pythag identities using tan, so i squared the first equation and solved for [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Square both sides to give [unparseable or potentially dangerous latex formula]. This means that [unparseable or potentially dangerous latex formula]. Luckily for us by inspection we find that [unparseable or potentially dangerous latex formula] is a root, so this factors into [unparseable or potentially dangerous latex formula]

We know that [unparseable or potentially dangerous latex formula] because that would make [unparseable or potentially dangerous latex formula]. This means that [unparseable or potentially dangerous latex formula]. Using the magical squaring ability gives [unparseable or potentially dangerous latex formula]**

We are trying to find [unparseable or potentially dangerous latex formula]. Substituting the values for [unparseable or potentially dangerous latex formula] yields [unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]

**Difference of two squares [unparseable or potentially dangerous latex formula]. So [unparseable or potentially dangerous latex formula]