2015 USAMO Problems/Problem 4
Problem
Find all functions such that
for all rational numbers
that form an arithmetic progression. (
is the set of all rational numbers.)
Solution
According to the given, , where
and
are rational. Likewise,
. Hence
, namely
. Let
, then consider
, where
and
. We have:
By induction,
for all in.tegers
.
Therefore, for nonzero integer
,
, namely
.
Hence
. Letting
, we obtain
, where
is the slope of the linear functions, and
.