1. A tractor is pulling a very long pipe on sled runners. Gavrila walked along the entire pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters. The speed of the tractor is constant.
Answer: 108 m. Solution. Let the length of the pipe be $x$ (meters), and for each step Gavrila takes of length $a$ (m), the pipe moves a distance of $y$ (m). Then, if $m$ and $n$ are the number of steps Gavrila takes in one direction and the other, respectively, we get two equations: ${ }^{x=m(a-y)}, x=n(a+y)$. From this, $\frac{x}{m}+\frac{x}{n}=2 a, \quad x=\frac{2 a m n}{m+n}$. With the given numerical values, we get $x=\frac{2 \times 0.8810160}{210+100}$ $=\frac{3360}{31} \approx 108.387$ (m).

Other methods of solving are possible. For example: With the same unknowns as above, let's also introduce Gavrila's speed $V$ (m/min) and the tractor's speed $U$ (m/min). Recording the time in both cases, we get the system:

$$
\left\{\begin{array}{l}
\frac{x}{V-U}=\frac{a m}{V} \\
\frac{x}{V+U}=\frac{a n}{V}
\end{array} \Leftrightarrow \left\{\begin{array}{l}
\frac{x}{a m}=1-\frac{U}{V} \\
\frac{x}{a n}=1+\frac{U}{V}
\end{array}\right.\right.
$$

Adding these, we get the same answer $x=\frac{2 a m n}{m+n}$.

Answer to variant 192: 98 m. Comment. $\frac{1463}{15} \approx 97.533$ (m).

Answer to variant 193: 89 m. Comment. $\frac{1152}{13} \approx 88.615$ (m).

Answer to variant 194: 82 m. Comment. $\frac{1071}{13} \approx 82.3846$ (m).