Easy as... 3.14?
by shiningsunnyday, Dec 3, 2016, 2:57 PM
Combo 3 homework (Medium) wrote:
Let
be positive integers. Prove that 


No point in dealing with all the prime factors at once. Define the p-adic map:
so essentially our problem reduces to 
In less formal terms, for every
element subset of
we take its smallest element, and we add it to our total if
is odd and subtract from our total if
is even, and we want to prove after doing this for all subsets we end up with the biggest element of 
WLOG let
Now consider how many times
is added in the right hand side. If we can show it is added a net total of zero times for
but just once when
our life is complete.
Consider the elements
In the first summation,
is obviously added
time. In the second summation, it's subtracted
times, once every time
is compared to one of
Similarly, the number of times it's added is
times, which is equal to
iff
and zero otherwise. And we're done!


In less formal terms, for every





WLOG let




Consider the elements









Tidbit
Don't even have time to attempt the harder and more interesting problems cause of school work. >.<
One thing I realized is that the Combo 3 problems are of good quality so I should probably run through the harder ones during winter break.
Ok back to the torture chamber.
One thing I realized is that the Combo 3 problems are of good quality so I should probably run through the harder ones during winter break.
Ok back to the torture chamber.