From point $M$, an ant crawls along a plane at a constant speed. Its path is a spiral that winds around point $O$ and is homothetic to some part of itself relative to this point. Will the ant be able to complete its entire path in a finite time?
An ant travels the entire path along a spiral from $M$ to $O$ in a finite time. Indeed, if the homothety coefficient $k<1$, and the ant spends time $t$ on the first turn, then on the remaining turns it will spend time $k t, k^{2} t$, $k^{3} t, \ldots$; therefore, the time for the entire path is $t\left(1+k+k^{2}+\ldots\right)=\frac{t}{1-k}$.