## Balkan Mathematica Olympiad 2017 – Problem 3

**Problem 3.** Let denote the set of positive integers. Find all functions such that

for all

*Solution:* Note that we obviously have

and using the hypothesis we obtain that

for every . Now we have two options. Suppose the image of is unbounded. Then, since if we fix it would follow that has arbitrarily large divisor, which is not possible unless . This is one solution, as it can easily be checked.

The other alternative is that is bounded. If this is true, then there is a value of which is repeated for an infinite increasing sequence : . It follows that

for every and for defined as above. Since the fraction

is an integer and it converges to as it follows that for large enough we have . This implies that , and this is independent of . Therefore is constant. From the previous relation we have so the only possible constant is .