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AIME #8 2004
Posted:
Sun Apr 22, 2007 11:23 pm
by stupidityismygam
How many positive integer divisors of [unparseable or potentially dangerous latex formula] are divisible by exactly 2004 positive integers?
try owning this problem quell
Posted:
Sun Apr 22, 2007 11:31 pm
by Quelloquialism
Um.
3. And a half.
Posted:
Sun Apr 22, 2007 11:40 pm
by Englandude
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.
Posted:
Sun Apr 22, 2007 11:43 pm
by Quelloquialism
No, it's 54.
Posted:
Mon Apr 23, 2007 12:31 am
by Quelloquialism
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.
Posted:
Mon Apr 23, 2007 10:07 am
by Michael T
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?
Posted:
Mon Apr 23, 2007 10:09 am
by bradp
AIME gives points based on the answer only... USAMO+ is proof writing
Posted:
Mon Apr 23, 2007 10:13 am
by Michael T
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.
Posted:
Mon Apr 23, 2007 10:14 am
by bradp
make it to USAMO then worry about how hard it is
Posted:
Mon Apr 23, 2007 10:16 am
by Michael T
(I hate IE, btw. If flood control owns you it deletes what you typed.)
Too true. I fail at this stuff right now.
Posted:
Mon Apr 23, 2007 10:17 am
by bradp
we're in the same boat.
Posted:
Mon Apr 23, 2007 10:46 am
by zefuri
USAMO is DEFINITELY harder
Posted:
Mon Apr 23, 2007 5:17 pm
by zefuri
amc's and aime are all pretty easy...