Maths Olympiad Prep

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Combinatorics Difficulty 5.5 AIME, harder Find the answer Netherlands

There are 20232023 people sitting at a round table. Each person is either a knave or a knight. Knights always speak the truth and knaves always lie. The first person says: "There is at least one knave at this table." The person to the left says: "There is at least one knight at this table." The third one says: "There are at least two knaves at this table." The fourth says: "There are at least two knights at this table." This continues, until the last person at the table says: "There are at least 10121012 knaves at this table." The first person, who has already made a statement, now says: "There are at least 10121012 knights at this table."
How many knaves are sitting at the table?
A) 505505 B) 506506 C) 10111011 D) 10121012 E) 15071507

This was a multiple-choice question, but the options didn't survive into the source we have. The answer given is B, and the solution below works it through.

Solution

B) 506506

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