Question 10
A4 (GBR). The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: For every $n\ge0$,
(i) $F(4n)=F(2n)+F(n)$;
(ii) $F(4n+2)=F(4n)+1$;
(iii) $F(2n+1)=F(2n)+1$.
Prove that for each positive integer $m$, the number of integers $n$ with $0\le n\lt 2^m$ and $F(4n)=F(3n)$ is $F(2^{m+1})$.
(i) $F(4n)=F(2n)+F(n)$;
(ii) $F(4n+2)=F(4n)+1$;
(iii) $F(2n+1)=F(2n)+1$.
Prove that for each positive integer $m$, the number of integers $n$ with $0\le n\lt 2^m$ and $F(4n)=F(3n)$ is $F(2^{m+1})$.