Question 6
For a positive integer $n$, let $v_2(n)$ be the greatest integer $k\geq0$ such that $2^k\mid n$. Suppose $f:\mathbb N\to\mathbb N$ satisfies:
(i) $f(x)\leq3x$ for all $x\in\mathbb N$;
(ii) $v_2(f(x)+f(y))=v_2(x+y)$ for all $x,y\in\mathbb N$.
Prove that for every positive integer $a$ there is exactly one positive integer $x$ such that $f(x)=3a$.
(i) $f(x)\leq3x$ for all $x\in\mathbb N$;
(ii) $v_2(f(x)+f(y))=v_2(x+y)$ for all $x,y\in\mathbb N$.
Prove that for every positive integer $a$ there is exactly one positive integer $x$ such that $f(x)=3a$.