Valhalla Legends Forums Archive | Yoni's Math Forum | A Problem of the Day Series?

AuthorMessageTime
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
November 19, 2005, 4:45 PM
Ender
hrm, seeing how no one has responded, it doesn't seem like this idea is very popular...
November 20, 2005, 7:33 PM
Rule
Go ahead and post interesting problems.  It seems that anyone who comes up with the idea of "daily problems" never follows through.
November 22, 2005, 5:50 AM
JoeTheOdd
2+2=
November 23, 2005, 1:14 PM
shout
[quote author=Joe link=topic=13288.msg135128#msg135128 date=1132751670]
2+2=
[/quote]
5
...
Go Radiohead!
November 23, 2005, 6:58 PM
Ender
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 / log[sub]2[/sub]100! + 1 / log[sub]3[/sub]100! + 1 / log[sub]4[/sub]100! + ... + 1 / log[sub]100[/sub]100!
November 23, 2005, 7:07 PM
Yoni
[quote author=Shout link=topic=13288.msg135156#msg135156 date=1132772293]
[quote author=Joe link=topic=13288.msg135128#msg135128 date=1132751670]
2+2=
[/quote]
5
...
Go Radiohead!
[/quote]
Radiohead?!
November 24, 2005, 6:22 AM
shout
[quote author=Yoni link=topic=13288.msg135202#msg135202 date=1132813334]
[quote author=Shout link=topic=13288.msg135156#msg135156 date=1132772293]
[quote author=Joe link=topic=13288.msg135128#msg135128 date=1132751670]
2+2=
[/quote]
5
...
Go Radiohead!
[/quote]
Radiohead?!
[/quote]

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

[quote author=Ender link=topic=13288.msg135158#msg135158 date=1132772822]
1 / log[sub]2[/sub]100! + 1 / log[sub]3[/sub]100! + 1 / log[sub]4[/sub]100! + ... + 1 / log[sub]100[/sub]100!
[/quote]

It's 1. I'm guessing.

Btw, does the log[sub]100[/sub]100 include the '!'? I guess it does not make a diffrence though... log[sub]100[/sub]100! = 1! = 1...
November 24, 2005, 2:40 PM
Ender
The answer is 1, but log[sub]100[/sub]100! != 1.

log[sub]100[/sub]100! = 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[sub]x[/sub]100! 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[sub]b[/sub]x + log[sub]b[/sub]y= log[sub]b[/sub]xy    you can simplify the sequence to
log (2 * 3 * ... * 100) / log 100!
~ =1
November 24, 2005, 4:06 PM
Yoni
[quote author=Shout link=topic=13288.msg135206#msg135206 date=1132843205]
[quote author=Yoni link=topic=13288.msg135202#msg135202 date=1132813334]
[quote author=Shout link=topic=13288.msg135156#msg135156 date=1132772293]
[quote author=Joe link=topic=13288.msg135128#msg135128 date=1132751670]
2+2=
[/quote]
5
...
Go Radiohead!
[/quote]
Radiohead?!
[/quote]

Hail to the Theif, track one. The song is called 2+2=5.
[/quote]
Hail to the Thief: Released in 2003
Radiohead: Formed in 1989
George Orwell's 1984: Written in 1949
November 24, 2005, 6:42 PM
K
[quote author=Yoni link=topic=13288.msg135231#msg135231 date=1132857765]
George Orwell's 1984: Written in 1949
[/quote]

K's wallet
November 29, 2005, 5:46 AM
Yoni
:o
November 29, 2005, 6:03 AM

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