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 .