Question 8
For an integer $m$, denote by $t(m)$ the unique number in $\{1,2,3\}$ such that $m+t(m)$ is a multiple of $3$. A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $f(-1)=0$, $f(0)=1$, $f(1)=-1$ and
$$
f\left(2^{n}+m\right)=f\left(2^{n}-t(m)\right)-f(m) \quad \text{ for all integers } m, n \geq 0 \text{ with } 2^{n} \gt m.
$$
Prove that $f(3p) \geq 0$ holds for all integers $p \geq 0$.
$$
f\left(2^{n}+m\right)=f\left(2^{n}-t(m)\right)-f(m) \quad \text{ for all integers } m, n \geq 0 \text{ with } 2^{n} \gt m.
$$
Prove that $f(3p) \geq 0$ holds for all integers $p \geq 0$.