Maths Olympiad Prep

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Combinatorics Difficulty 6.0 AIME, harder Prove it Mongolia

nn girls are standing in a circle, each holding exactly 1 card. One girl gives her card to the girl on her left, who in turn gives 2 cards to the girl on her left. The girl who got the cards gives 1 card to the girl on her left. The girl who got the card gives 2 cards to the girl on her left. Continuing this way, each girl gives alternating 1 or 2 cards to the girl on her left. Anyone who has no card leaves the game immediately. Find all values of nn such that all cards are collected by one girl.

Solution

Answer: n=2n = 2, n=2k+1n = 2^k + 1, n=2k+2n = 2^k + 2, k≥1k \ge 1.
Solution omitted.

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