## Balkan Mathematical Olympiad 2012 Problem 4

Let be the set of positive integers. Find all functions such that the following conditions both hold:

(i) for every positive integer ,

(ii) divides whenever and are different positive integers.

*Balkan Mathematical Olympiad 2012 Problem 4*

**Solution: **First we have for , which means that . We have a few cases:

**Case 1: **. Then we know that and . This means that is odd for every and therefore for every ,

**Case 2: **. First note that so is even. Then which means that necessarily , and furthermore . This means that modulo , and this means that is never divisible by . Since is also even it follows that on .

**Case 3: **. It follows that has last digit and cannot be a factorial.

**Case 4: **. For we have and since the function is strictly increasing for we find that , and furthermore , which allows us to deduce from that .

So with . Then , which can hold only if . In this case we have .

where is the problem?

Do you mean where is the solution? I haven’t posted it yet.

I didn’t mean that.I’m very sorry Beni because I didn’t read the post carefully.The problem is to find all function f that satisfy the two condition isn’t it?

Yes, that is the problem 🙂