An airplane traces out a regular $n$-sided polygon, and does so at speeds of $v_{1}, v_{2}, \ldots v_{n}$ along each side. What is the average speed of the airplane?
If the side of the polygon is $a$, then the perimeter of the polygon the airplane writes down is

$$
t=\frac{a}{v_{1}}+\frac{a}{v_{2}}+\ldots+\frac{a}{v_{n}}
$$

with the average speed $v$, it is written down in

$$
t^{\prime}=\frac{n a}{v}
$$

time. Since $t^{\prime}=t$,

$$
\frac{1}{v}=\frac{1}{n}\left(\frac{1}{v_{1}}+\frac{1}{v_{2}}+\ldots+\frac{1}{v_{n}}\right)
$$

that is, the average speed $v$ is the harmonic mean of the speeds $v_{1}, v_{2}, \ldots, v_{n}$.

István Papp (State Fazekas Mihály Grammar School VII. o. Debrecen.)