A chess grandmaster has 11 weeks to prepare for the World Chess Championships. For training he decides to play at least one game every day. However, in order not to tire himself he decides not to play more than 12 games in any consecutive 7 day period.

Show that there exists a succession of days during while he plays *exactly* 22 games (No more, no less). Give proof.

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## Comments

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TopNewest(This solution is not fully original)\[\]Let us say that \(a_n\) denotes the number of games he plays in the first \(n\) days of his practice.Now let us consider the \(154\) numbers,\(a_1,a_2,...a_{76},a_{77},a_1+22,a_2+22,...a_{76}+22,a_{77}+22\), in this set the largest number is \(\leq 11 \times 12+22=154\),now since all the numbers \(a_i\) are different as he plays at least one game a day.So we see,by PHP,that if he doesn't play \(12\) games in any \(7\) day period then two numbers would be equal,hence it would mean that,\(a_k+22=a_i\) hence proving that he definitely played \(22\) games in a consecutive day period.But,if he plays \(12\) games every period of \(7\) days,then......

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Can you finish your solution please.

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Yes,sorry about that,have been a little busy this week.I don't think i will be much active till Dec 6th(RMO).

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Comment deleted Nov 13, 2015

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But the maximum number of games he will play is 12, and the minimum is 7. He might not play 12 games every consecutive 7 day.

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As mentioned by Julian, it is not given that he plays 12 games per week.

Even assuming that he plays a total of 132 games, it's possible that he plays 21, 23, 21, 23, 21, 23 games in the 6 periods that you defined, and hence there isn't an "exactly 22 games" during those periods.

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