▪ Romania District Olympiad 2012

1. Let three positive distinct real numbers. Evaluate:

.

2. Let a 9 elements ring. Prove that the following assertions are equivalent:

(a) For any there are two numbers and such that

  .

(b) is a field. 

3. Let a elements group. Find all the functions such that:

(a) if and only if is 's identity;

(b) for any divisor of , where stands for the greatest common divisor of the positive integers and . 

4. Let a differentiable function such that and . Prove that:

.