3. There are 48 matches. They are laid out in three unequal piles. Then three transfers of matches from pile to pile are performed. First, from the first pile to the second, as many matches are transferred as there were in the second pile. Then, from the second pile to the third, as many matches are transferred as there are in the third pile. Finally, from the third pile to the first, as many matches are transferred as there are in the first pile. As a result of these three transfers, the number of matches in all piles became equal. How many matches were there in each pile initially? (10 points)

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# Solution:

We solve the problem from the end. As a result of all three transfers, the number of matches in all piles became the same, i.e., 48: 3 = 16 matches in each pile. During the third transfer, the number of matches in the first pile was doubled by adding as many matches as it already had. Therefore, before the third transfer, there were 16: 2 = 8 matches in the first pile. In the third pile, from which 8 matches were taken, there were 16 + 8 = 24 matches before the third transfer. Thus, before the third transfer, the distribution of matches in the piles was as follows: the first pile - 8 matches, the second pile - 16 matches, the third pile - 24 matches. During the second transfer, the number of matches in the third pile was doubled by adding as many matches as it already had. Therefore, before the second transfer, there were 24: 2 = 12 matches in the third pile. In the second pile, from which 12 matches were taken, there were 16 + 12 = 28 matches before the second transfer. Thus, before the second transfer, the distribution of matches in the piles was as follows: the first pile - 8 matches, the second pile - 28 matches, the third pile - 12 matches. During the first transfer, the number of matches in the second pile was doubled by adding as many matches as it already had. Therefore, before the first transfer, there were 28: 2 = 14 matches in the second pile. In the first pile, from which 14 matches were taken, there were 8 + 14 = 22 matches before the first transfer. Thus, before the first transfer, i.e., initially, the distribution of matches in the piles was as follows: the first pile - 22 matches, the second pile - 14 matches, the third pile - 12 matches.

Answer: {22; 14; 12}.

| Points | Criteria for evaluating the performance of task No. 3 |
| :---: | :--- |
| **10** | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are correctly performed. The correct answer is obtained. |
| **8** | A correct sequence of all steps of the solution is provided. There are gaps in the justification of the solution or a computational error or typo that does not affect the further course of the solution. As a result of this error or typo, an incorrect answer may be obtained. |
| **5** | The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is performed correctly, possibly inaccurately (for example, the distribution of matches after a separate transfer is found); - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the distribution of matches after all transfers is found incorrectly). The solution may be incomplete. |
| **0** | All cases of the solution that do not correspond to the above criteria for scoring 5, 8, and 10 points. |