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How many positive integer divisors of [unparseable or potentially dangerous latex formula] are divisible by exactly 2004 positive integers?

try owning this problem quell

Um.

3. And a half.

Let me try and make an educated guess....how about 2004?? Am i rite? Am i rite?

I have no clue in other words. Hey, i put forth an "educated" effort.

No, it's 54.

Factor 2004.

You get [unparseable or potentially dangerous latex formula]. Distribute the exponent, and you have [unparseable or potentially dangerous latex formula].

The number of positive integral divisors of an integer is the product of all prime exponents increased by one.

So, we are looking for solutions to the equation [unparseable or potentially dangerous latex formula], where a, b, and c are all integers between 0 and 2003, inclusive (the 2003 limit is because, clearly, no solutions are possible if all terms are at least 1 and one term is at least 2003), representing the exponents of 167, 2, and 3, respectively.

As we already determined, 2004 = [unparseable or potentially dangerous latex formula].

Then my elegance engine crashed and I brute-forced the possible values of a, b, and c on paper, though I'm sure there's a better way. My brain doesn't work.

what's the point difference between bruteforcing and eleganc(ing?) Does it make a huge difference if you find a nice solution to a problem vs what quell usually does?

AIME gives points based on the answer only... USAMO+ is proof writing

but that's not to be confused with saying AIME is easier than USAMO, right?

The reason I've started browsing this forum is because I'm thinking about participating next year.

make it to USAMO then worry about how hard it is

(I hate IE, btw. If flood control owns you it deletes what you typed.)

Too true. I fail at this stuff right now.

we're in the same boat.

USAMO is DEFINITELY harder

amc's and aime are all pretty easy...