1. Petya ran down the escalator, counting the steps. Exactly halfway down, he stumbled and the rest of the way he tumbled down (Petya flies three times faster than he runs). How many steps are there on the escalator if Petya counted 20 steps with his feet (i.e., before falling) and 30 steps with his sides (after falling)?
Answer: 80

Solution. Let the escalator have a length of $2 \mathrm{~L}$ (steps), the speed of the escalator be $u$, and Petya runs with a speed of $\mathrm{x}$, and flies with a speed of $3 \mathrm{x}$. Then the time until the fall is $\frac{L}{u+x}$ and during this time Petya will count $\frac{L x}{u+x}$ steps. From this: $\frac{1}{20}=\frac{1}{L}\left(1+\frac{u}{x}\right)$. A similar equation is obtained "after the fall": $\frac{1}{30}=\frac{1}{L}\left(1+\frac{u}{3 x}\right)$. Multiplying by 3, and subtracting the first, we get: $\frac{3}{30}-\frac{1}{20}=\frac{2}{L}$, so $\mathrm{L}=40$, and $2 \mathrm{~L}=80$.

Grading. Full solution - 7 points.