Croatia_2018

Question

Let f: N N be a function such that f(ab) = f(a+b) for all positive integers a 4 and b 4. Prove that f(n) = f(8) for all positive integers n 8.

Accepted Answer

Let n 8 be a positive integer. The problem condition implies that aligned f(n) &= f(4 + (n-4)) = f(4(n-4)) = f(2(n-4) + 2(n-4)) &= f(4(n-4)(n-4)) = f(4(n-4) + (n-4)) &= f(5(n-4)) = f(5 + n - 4) &= f(n+1). aligned Therefore, by the principle of mathematical induction, we conclude that f(n) = f(8) for all n 8.