4. 20 balls of the same mass are moving along a chute towards a metal wall with the same speed. Coming towards them at the same speed are 16 balls of the same mass. When two balls collide, they fly apart with the same speed. After colliding with the wall, a ball bounces off it with the same speed. (The balls move only along the chute). How many collisions will there be between the balls?
# Answer: 510.

Solution. We will assume that initially, each ball moving towards the wall has a red flag, and the rest of the balls have blue flags. Imagine that when the balls collide, they exchange flags. Then each blue flag moves at a constant speed in one direction (away from the wall), and each red flag reaches the wall and then moves in the opposite direction. The number of collisions between the balls is equal to the number of flag exchanges. Each red flag will exchange once with each blue flag. Any two red flags will also exchange places once. Therefore, the total number of exchanges is $20 \cdot 16 + \frac{20 \cdot 19}{2} = 510$.

Evaluation. 14 points for a correct solution.

## $2015 / 16$ academic year

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