## The 41-st Tournament of the Towns

Today I present three problems from the 41-st Tournament of the Towns that I liked: an easy one, one that reminds me of the Collatz conjecture, and a hard one.

Problem 1 (by Aleksey Voropayev).A magician places all the cards from the standard 52-card deck face up in a row. He promises that the card left at the end will be the ace of clubs. At any moment, an audience member tells a numbernthat doesn’t exceed the number of cards left in the row. The magician counts thenth card from the left or right and removes it. Where does the magician need to put the ace of clubs to guarantee the success of his trick?

Problem 2 (by Vladislav Novikov).Numberxon the blackboard can be replaced by either 3x+ 1 or ⌊x/2⌋. Prove that you can use these operations to get to any natural number when starting with 1.

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Problem 3 (by A. Gribalko).There are 2nconsecutive integers written on a blackboard. In one move, you can split all the numbers into pairs and replace every paira,bwith two numbers:a+banda−b. (The numbers can be subtracted in any order, and all pairs have to be replaced simultaneously.) Prove that no 2nconsecutive integers will ever appear on the board after the first move.

## Sanjay:

Either end of the row of 52 cards.

30 December 2021, 1:59 am