5. Gavriil got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was completely drained. For half of the time, he played Tetris, and for the other half, he watched cartoons. It is known that the smartphone fully discharges in 3 hours of video watching or in 5 hours of playing Tetris. What distance did Gavriil travel if the train moved half the distance at an average speed of 80 km/h and the other half at an average speed of 60 km/h? Give the answer in kilometers, rounding to the nearest whole number if necessary.
Answer: 257

Let's assume the "capacity" of the smartphone battery is 1 unit (u.e.). Then the discharge rate of the smartphone when watching videos is $\frac{1}{3}$ u.e./hour, and the discharge rate when playing games is $\frac{1}{5}$ u.e./hour. If the total travel time is denoted as $t$ hours, we get the equation $\frac{1}{3} \cdot \frac{t}{2} + \frac{1}{5} \cdot \frac{t}{2} = 1$, which simplifies to $\frac{(5+3) t}{2 \cdot 3 \cdot 5} = 1$, or $t = \frac{15}{4}$ hours. Then we have (where $S$ is the distance): $\frac{S}{2 \cdot 80} + \frac{S}{2 \cdot 60} = \frac{15}{4}$, which simplifies to $\frac{S}{40} + \frac{S}{30} = 15$,

$$
S = 15 \cdot \frac{40 \cdot 3}{4+3} = \frac{1800}{7} \approx 257 \text{ km. }
$$

## Lomonosov Olympiad in Mechanics and Mathematical Modeling - 2021/2022

9th grade

Each problem is worth 20 points. A score of 20 points is given for a correct and complete solution and the correct answer.

A score of 15 points is given for a solution with various deficiencies (inadequate justification, inaccuracies, etc.). In some problems, scores of 5 and 10 points were also given for partial progress in the solution.

## Grading Criteria for Problems: