IMO Shortlist 1984 problem 16
Dodao/la:
arhiva2. travnja 2012. Let
![a,b,c,d](/media/m/7/6/0/7605ede133e1f767d3890e0bfffb7b7f.png)
be odd integers such that
![0<a<b<c<d](/media/m/b/2/f/b2fc570db4be2d68dfd44fc54de68f20.png)
and
![ad=bc](/media/m/b/4/4/b448a61fdea610cae5b4bf692b962490.png)
. Prove that if
![a+d=2^k](/media/m/8/1/7/817bb45f82205eac059df0e0189a8e1c.png)
and
![b+c=2^m](/media/m/7/6/d/76dbb8117b56786512863db7ade44b2d.png)
for some integers
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
and
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
, then
![a=1](/media/m/1/c/6/1c6abdce7cd19174d88d7aa73e680bf7.png)
.
%V0
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
Izvor: Međunarodna matematička olimpijada, shortlist 1984