Valhalla Legends Archive

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

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

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

2+2=

Quote from: Joe on November 23, 2005, 07:14 AM
2+2=

5
...
Go Radiohead!

Hm, I'll post a problem and see how enthusiastic you all are. We'll decide how to go from there :P
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!

Quote from: Shout on November 23, 2005, 12:58 PM

Quote from: Joe on November 23, 2005, 07:14 AM
2+2=

5
...
Go Radiohead!

Radiohead?!

Quote from: Yoni on November 24, 2005, 12:22 AM

Quote from: Shout on November 23, 2005, 12:58 PM

Quote from: Joe on November 23, 2005, 07:14 AM
2+2=

5
...
Go Radiohead!

Radiohead?!

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

Quote from: Ender on November 23, 2005, 01:07 PM
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...

The answer is 1, but log₁₀₀100! != 1.

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

so show your work :P

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

Quote from: Shout on November 24, 2005, 08:40 AM

Quote from: Yoni on November 24, 2005, 12:22 AM

Quote from: Shout on November 23, 2005, 12:58 PM

Quote from: Joe on November 23, 2005, 07:14 AM
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

Quote from: Yoni on November 24, 2005, 12:42 PM
George Orwell's 1984: Written in 1949

K's wallet (http://ucsu.colorado.edu/~ledbettj/LoveIsHate.png)

:o