1983 IMO Problems/Problem 1
Find all functions defined on the set of positive reals which take positive real values and satisfy:
for all
; and
as
.
Let and we have
. Now, let
and we have
which simplifies to
and since
we have
.
Plug in and we have
. If
is the only solution to
then we have
. We prove that this is the only function by showing that there does not exist any other
:
Suppose there did exist such an . Then, letting
in the functional equation yields
. Then, letting
yields
. Notice that since
, one of
is greater than
. Let
equal the one that is greater than
. Then, we find similarly (since
) that
. Putting
into the equation, yields
. Repeating this process we find that
for all natural
. But, since
, as
, we have that
which contradicts the fact that
as
.