IMO 2013 #1
by Wolstenholme, Oct 27, 2014, 2:14 PM
Prove that for any pair of positive integers 
and 
, there exist positive integers 
(not necessarily different) such that
![\[ 1+\frac{2^k-1}{n}=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\dots\left(1+\frac{1}{m_k}\right). \]](//latex.artofproblemsolving.com/f/d/2/fd22e2287c1f0f11d08c1a01338dbbb565af4e2d.png)
Proof:
The result follows immediately from induction and the identities
and 
.
The motivation is pretty natural - just looking at the problem one immediately sees that induction is the way to go and after writing out a few small cases the pattern becomes evident.
and 
, there exist positive integers 
(not necessarily different) such that![\[ 1+\frac{2^k-1}{n}=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\dots\left(1+\frac{1}{m_k}\right). \]](http://latex.artofproblemsolving.com/f/d/2/fd22e2287c1f0f11d08c1a01338dbbb565af4e2d.png)
Proof:
The result follows immediately from induction and the identities
and 
.The motivation is pretty natural - just looking at the problem one immediately sees that induction is the way to go and after writing out a few small cases the pattern becomes evident.
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