5. Around a round table, there are 11 chairs, and at the beginning, 10 children are sitting whose chairs, in the direction of the clock hands, are marked with numbers from 1 to 10. Pavel needs to join them. Once everyone is seated around the table, the children play an elimination game: they start counting in the direction of the clock hands from the child sitting on the chair marked with number 1, and the thirteenth child is eliminated. The remaining children continue the game from where the counting stopped, the thirteenth again is eliminated, and so on, until only one child remains. At which place, or between which two of the ten children who were already sitting at the table, should Pavel sit to win the game?
Solution. When Pavel sits down, there will be 11 children at the table. We renumber the chairs around the table in the direction of the clock hands, so that the chair previously marked with the number 1 is again marked with the number 1. We start counting as described in the game and cross out every thirteenth number. We sequentially cross out $2,5,9,4,1,3,8,10,11,6$, and at the end, the number 7 will remain, which means that Pavel should sit between the children who are sitting on the chairs originally marked with the numbers 6 and 7.

## VIII Department