Problem 1. A train consists of 20 cars, numbered from 1 to 20, starting from the beginning of the train. Some of the cars are postal cars. It is known that

- the total number of postal cars is even;
- the number of the nearest postal car to the beginning of the train is equal to the total number of postal cars;
- the number of the last postal car is four times the number of postal cars;
- any postal car is coupled with at least one other postal car.

Find the numbers of all postal cars in the train.
Answer: $4,5,15,16$.

Solution. The number of the last postal car cannot exceed 20; the number of postal cars is four times less than this number and therefore does not exceed 5. Since the number of postal cars is even, there are 2 or 4 of them.

Suppose there are two. Then the first one has the number 2, and the last one has the number 8. But they are not adjacent, which contradicts the condition.

Therefore, there are four postal cars. The first one has the number 4; another postal car is coupled with it - it can only have the number 5. The last postal car has the number 16; the one coupled with it has the number 15.

## Criteria

Any complete solution to the problem is worth 7 points. In the absence of such a solution, the following criteria are summed up:

3 p. Correct answer.

2 p. Proven that there cannot be 6 or more postal cars.

2 p. Proven that there cannot be 2 postal cars.