- Let be the greatest odd divisor of , show that exists and find it (
Bulgaria 1985)- Find all such that , (
Olimpíada Rioplatense 2008) - is defined as in 1 -

Have fun! (Happy)

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- January 5th 2009, 02:53 PMPaulRS2 interesting problems
- Let be the greatest odd divisor of , show that exists and find it (
**Bulgaria 1985**) - Find all such that , (
**Olimpíada Rioplatense 2008**) - is defined as in 1 -

Have fun! (Happy) - Let be the greatest odd divisor of , show that exists and find it (
- January 5th 2009, 04:39 PMNonCommAlg
- January 5th 2009, 06:56 PMchiph588@
- January 6th 2009, 01:34 AMPaulRS
First note that: (sum on both sides)

The idea is that iff

Indeed, to prove this:

, conversly if then and

Now since

It follows easily that: (only one of the terms is not 0 )

My proof of**(1)**is based on the following observation: then by a simple counting argument ( appears as many times as multiples of there are in between 1 and n -both included-, that is ): ... - January 7th 2009, 06:13 PMchiph588@
Sorry to be a bother, but I'm still a bit confused... (Wondering)

- May 30th 2010, 05:21 PMchiph588@
- June 1st 2010, 09:01 AMgmatt
Correct me if I am wrong but the question is not clear as stated. Does the question mean to ask

,

or perhaps

,

or does it mean to ask

,

the way that for all is used currently doesn't make sense to me, since the quantified variable is used to define the domain set . - November 13th 2010, 01:33 PMPaulRS
The bar there ("/") means "

*such that*". So what I mean is that that should hold for all integers n greater than 1.