Let
be relatively prime positive integers. Define the weight of an integer
, denoted by
to be the minimal possible value of
taken over all pairs of integers
and
such that
. An integer
is called a local champion if
and
. Find all local champions and determine their number.
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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.
Školjka