2015 USAJMO 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 1
According to the given, , where x and a are rational. Likewise
. Hence
, namely
. Let
, then consider
, where
.
,
.
Easily, by induction,
for all integers
.
Therefore, for nonzero integer m,
, namely
Hence
. Let
, we obtain
, where
is the slope of the linear functions, and
.
Solution 2
We have and
Subtracting these two and rearranging gives
and since
we get
from which we get
Then we have
. Setting
, we let
to get
. This is Cauchy's functional equation, so it has solutions at
, so the answer is
.