1. A clown is riding a bicycle around the circus arena, where the radius of the circumference of the front wheel is half the radius of the circumference of the rear wheel. If the circumference of the front wheel were increased by a meter, and the circumference of the rear wheel were decreased by a meter, then over 40 meters, the rear wheel would make 20 more revolutions than the front wheel. Determine the lengths of the circumferences of the wheels.
Solution. The circumference of the front wheel $C_{n}=2 \pi R_{n}$, the rear - $C_{z}=2 \pi R_{z}=2 \pi 2 R_{n}$, the number of revolutions after changes in the circumferences are related by

$$
\begin{aligned}
& \frac{40}{C_{z}-1}-20=\frac{40}{C_{n}+1} \Rightarrow \frac{2}{2 C_{n}-1}-1=\frac{2}{C_{n}+1} \Rightarrow 2 C_{n}^{2}+3 C_{n}-5=0 \Rightarrow C_{n}=\left[\begin{array}{l}
1 \\
-10 / 4<0
\end{array}\right. \\
& \Rightarrow C_{n}=1, C_{z}=2
\end{aligned}
$$

Answer: $1 ; 2$.