Solution:

Let $w_{0}=0$, let $w_{r}$ be the vector $w$ after $r$ rounds. Also, let $e_{1}$ denote the vector $(1,0, \ldots, 0)$. Note that
$$w_{r+1}^{2}=\left(w_{r}+v_{i}\right)^{2}=w_{r}^{2}+v_{i}^{2}+2 w_{r} \cdot v_{i} \leq w_{r}^{2}+v_{i}^{2}$$
as we have by the condition that $w_{r} \cdot v_{i} \leq 0$. Therefore, we have by induction that $\left|w_{r}\right| \leq M \sqrt{r}$, where $M=\max _{i}\left|v_{i}\right|$. Also, we have that
$$w_{r+1} \cdot e_{1}=w_{r} \cdot e_{1}+v_{i} \cdot e_{1}$$
so $w_{r} \cdot e_{1} \geq m r$, where $m=\min _{i}\left(v_{i} \cdot e_{1}\right)>0$.
Now using the obvious inequality
$$M \sqrt{r} \geq\left|w_{r}\right| \geq w_{r} \cdot e_{1} \geq m r$$
we get that $r \leq \frac{M^{2}}{m^{2}}$, as desired.