Let $\mathbb{N}_{\ge 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$:
(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$,
(b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$,
(c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.
That's one of the nicest and most interesting functional equations of the last three years! One solution is $f(n)=n+1$. For the other solution you have to write $n=2^a*b$ with $b$ odd, and set $f(n)=2^a+b$.
This problem is artificial and contrived! Does it make sense adding the power of $2$ and the odd part?? It was constructed base on a fancy solution and added tediously long and ugly conditions. Why? In real life mathematics the solution $f(n)=2^a+b$ is meaningless and inspired nothing. Is this artificial solution just constructed to make the problem look more interesting?
We all must be regretful to the PSC that they have ruined Francophone MO with such an unnatural and artificial creature!