AIME II #7 2000

[stupidityismygam]

i guess ill post another one since nobody liked the other one

Given that [unparseable or potentially dangerous latex formula] find the greatest integer that is less than [unparseable or potentially dangerous latex formula]

[bradp]

lcm of the denominators is probably something like the denominator of the right side, [unparseable or potentially dangerous latex formula]

i'll leave the grunt work to someone else

[mathpimp]

Alright, mine's pretty much using brute force too, and I didn't write much out either so I'll just kinda summarize.

Multiply over the [unparseable or potentially dangerous latex formula] and distribute it. Then of course you're cancelling out a bunch of stuff until you get your sequence of numbers that you add together.

Finally the easy part:

[unparseable or potentially dangerous latex formula]

Those are all the remaining numbers multiplied together after cancellations. I didn't have a calculator so I wrote them down that way.

Basically that means [unparseable or potentially dangerous latex formula].

[unparseable or potentially dangerous latex formula]

Greatest integer less than that is obviously [unparseable or potentially dangerous latex formula].

[stupidityismygam]

you fail at addition the answer is 137

Super Duper awesome solution power go...

The statement given looks a lot of combinations, so to exploit this similarity we will multipky both sides by [unparseable or potentially dangerous latex formula]. This gives

[unparseable or potentially dangerous latex formula]

Thus,
[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

So then with more magicalness, we get [unparseable or potentially dangerous latex formula]

Which means[unparseable or potentially dangerous latex formula]

Divide by 19 to give, [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

So [unparseable or potentially dangerous latex formula]

[mathpimp]

Wow man, that required a lot more thought then what I did with the retarded algebra.

Anyway, interestingly enough, by the time I actually found a calculator the only thing I used it for was the final addition! So in other words I # at typing stuff in the calculator. All the stuff I did mentally was right because I was more cautious I guess.

So, this further confirms that its good that I never did Calculator UIL.

[mathpimp]

Oh, yeah I see your point.

nCk=nC(n-k)