A Problem of the Day Series?

[Ender]

How would you guys like this?

EDIT

I know there are lots of other sites/forums that do this, but it would be nice to have it within this community =p

[Ender]

hrm, seeing how no one has responded, it doesn't seem like this idea is very popular...

[Rule]

Go ahead and post interesting problems.  It seems that anyone who comes up with the idea of "daily problems" never follows through.

[Joe[x86]]

2+2=

[shout]

2+2=

5
...
Go Radiohead!

[Ender]

Hm, I'll post a problem and see how enthusiastic you all are. We'll decide how to go from there
Show your work as well as your answer.

* Taken from AoPS, which got this problem from AHSME 1998 #22

Find the value of the expression:

1 / log₂100! + 1 / log₃100! + 1 / log₄100! + ... + 1 / log₁₀₀100!

[Yoni]

2+2=

5
...
Go Radiohead!

Radiohead?!

[shout]

2+2=

5
...
Go Radiohead!

Radiohead?!

Hail to the Theif, track one. The song is called 2+2=5.

1 / log₂100! + 1 / log₃100! + 1 / log₄100! + ... + 1 / log₁₀₀100!

It's 1. I'm guessing.

Btw, does the log₁₀₀100 include the '!'? I guess it does not make a diffrence though... log₁₀₀100! = 1! = 1...

[Ender]

The answer is 1, but log₁₀₀100! != 1.

log₁₀₀100! = log 100! / log 100  = (log 100 + log 99 + ... + log 1) / log 100.

so show your work

EDIT

once again, judging from the replies, i see this isn't very popular =[

EDIT2

ok seeing how no one cares about this, I'll just post the work to get the answer:

~ first step is to get a common denominator
~ substitute x as the base, the terms in the sequence will now look like 1 / log_x100! where x is the base.
~ change-a-base to get 1 / log 100! / log x  = log x / log 100!               
~ plug in the bases for x, rewrite the sequence: log 2 / log 100! + log 3 / log 100! + ... + log 100 / log 100!
~ simplify: (log 2 + log 3 + ... + log 100)/log 100!   b/c of common denominator
~ using properties of logs, log_bx + log_by= log_bxy     you can simplify the sequence to
log (2 * 3 * ... * 100) / log 100!
~ =1

[Yoni]

2+2=

5
...
Go Radiohead!

Radiohead?!

Hail to the Theif, track one. The song is called 2+2=5.

Hail to the Thief: Released in 2003
Radiohead: Formed in 1989
George Orwell's 1984: Written in 1949

[K]

George Orwell's 1984: Written in 1949

K's wallet

[Yoni]