1990 OIM Problems/Problem 1
Problem
Let be a function defined in the set of integers greater or equal to zero such that:
(i) If , for all then
(ii) If , for all then
a. Prove that for all integer , greater or equal to zero, there exist an integer grater than zero such that
b. Calculate .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Part a.
, , , and so on..
So we pick a range where which is where is a non-negative integer.
Therefore, which provides the range for as:
In that range, this gives
When , which proves the equality since .
When , which also proves the equality since .
Part b.
- This seems to be one of the easiest problems I've seen in these types of competitions, unless I'm missing something.
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.