1. The brother left the house 6 minutes after his sister, following her, and caught up with her after 12 minutes. How many minutes would it have taken him to catch up if he had walked twice as fast? Both the brother and the sister walk at a constant speed.
# Answer: 3 minutes.

Solution. Since the brother walked for 12 minutes before meeting his sister, and the sister walked for 18 minutes, the brother's speed was $3 / 2$ times the sister's speed. If the brother's speed is 3 times the sister's speed, which is 2 times faster than before, then the difference of 6 minutes will be covered 4 times faster than before, that is, in 3 minutes.

## Grading Criteria:

- The correct answer with a complete justification is given - 7 points;
- The correct answer and a generally correct justification, with gaps or inaccuracies, is given - 5 points;
- The correct answer with verification is given, but the uniqueness is not proven - no more than 3 points;
- The correct answer is given without justification - 1 point;
- The incorrect answer is given - 0 points.