Buses are traveling on the highway, in both directions, at equal intervals and at equal speeds. A cyclist traveling at $16 \mathrm{~km} /$ hour starts counting the buses from the moment two buses meet next to him. The 31st bus coming from the opposite direction and the 15th bus from behind coincidentally met the cyclist again at the same moment.

How fast were the buses traveling?
I. Solution: If the cyclist is moving, the speed of the cars coming from the opposite direction appears to increase by the cyclist's speed, while from behind it appears to decrease by the same amount. In this ratio, the cyclist encounters more buses than he overtakes, so: $\frac{31}{c+16}=\frac{15}{c-16}$. From this, $31 \cdot(c-16)=15 \cdot(c+16)$, hence $c=46 \mathrm{~km} /$ hour.

Vígh Magda (Bp.-i Veres Pálné lg. V. a) o.)

Solved by: Fried E., Gehér L., Glatz J., Gombócz K., Gősy S., Horváth Sz., Róna P., Tihanyi L., Turczi Gy., Vermes R.

II. Solution: Consider the two buses with which the cyclist last meets, starting from the time he begins counting the buses. Let the distance between two adjacent buses be $d$, then the bus coming from the opposite direction and the cyclist together cover $31 d$ distance until they meet, while the bus coming from behind starts $15 d$ distance from the cyclist's starting point. If $A$ is the distance covered by the buses, and $K$ is the distance covered by the cyclist, then:

$$
A+K=31 d \quad \text { and } \quad A-K=15 d, \quad \text { so } \quad A=23 d, \quad K=8 d .
$$

Thus, the bus travels $\frac{23}{8}$ times faster than the bicycle.

Its speed is $16 \cdot \frac{23}{8}=46 \mathrm{~km} /$ hour.

Réthy Eszter (Bp.-i Veres Pálné lg. V. a) o.)

Solved by: Berendik I., Bognár J., Csillik B., Fehér S., Gacsányi S., Haris B., Kovács G., Kővári T., Neszményi A., Perjes P., Róna P., Seregély Gy., Szépfalussy P., Tarnay Gy., Turczi Gy., Vörös M., Zsigmond V.

Did not justify the calculation: Bagi Katalin, Bottlik I., Csernók L., Ember Gy., Horváth Zsuzsanna, Kalló D., Kiss J., Missura P., Mocsár Erzsébet, Retkes Margit, Székely Éva, Tarnóczi T., Tóth K., Vízi Mária.