1969 Canadian MO Problems/Problem 8
Problem
Let be a function with the following properties:
1) is defined for every positive integer
;
2) is an integer;
3) ;
4) for all
and
;
5) whenever
.
Prove that .
Solution
It's easily shown that and
. Since
Now, assume that is true for all
where
It follows that Hence,
, and by induction