IMO Shortlist 2006 problem N6
Dodao/la:
arhiva2. travnja 2012. Let
![a > b > 1](/media/m/9/9/3/9935aaad6321064109d4b8a1a7e2fc3f.png)
be relatively prime positive integers. Define the weight of an integer
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
, denoted by
![w(c)](/media/m/e/9/9/e990f57f8e04a16033b3dd822cee96e8.png)
to be the minimal possible value of
![|x| + |y|](/media/m/a/9/8/a98f8179fb3895d819f0ae516c0c8446.png)
taken over all pairs of integers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
such that
![ax + by = c](/media/m/2/f/e/2fe2b3c7622508649b2889979e7e6796.png)
. An integer
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
is called a local champion if
![w(c) \geq w(c \pm a)](/media/m/3/d/f/3df008e89fba9b4ed21d29615b3ef1c7.png)
and
![w(c) \geq w(c \pm b)](/media/m/0/5/c/05cb6504671a768e97c53b2f6300d084.png)
. Find all local champions and determine their number.
%V0
Let $a > b > 1$ be relatively prime positive integers. Define the weight of an integer $c$, denoted by $w(c)$ to be the minimal possible value of $|x| + |y|$ taken over all pairs of integers $x$ and $y$ such that $ax + by = c$. An integer $c$ is called a local champion if $w(c) \geq w(c \pm a)$ and $w(c) \geq w(c \pm b)$. Find all local champions and determine their number.
Izvor: Međunarodna matematička olimpijada, shortlist 2006