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[联谊] Putnam Contest problems.

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发表于 2002-12-7 23:45 | 显示全部楼层 |阅读模式
went to take the Putnam Math Contest today. here are 4 questions from it (not the most difficult ones). everybody is welcome to try them and share your excellent ideas. university freshman level math is more than enough.

Question 1:
An integer n, unknown to you, has been randomly chosen in the interval [1,2002] with uniform probability. your objective is to select n in an  odd  number of guesses. after each incorrect guess, you are informed whether n is higher or lower, and you  must  guess an integer on your next turn among the numbers that are still feasibly correct. show that you have a stratedy so that the chance of winning is greater than 2/3.
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 楼主| 发表于 2002-12-8 14:16 | 显示全部楼层
Question 2:

let n>=2 be an integer and Tn be the number of non-empty subsets S of {1,2,3,...,n} with the property that the average of the elements of S is an integer. prove that Tn - n is always even.
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 楼主| 发表于 2002-12-8 14:22 | 显示全部楼层
Question 3:

in determinant Tic-Tac-Toe, player 1 enters a 1 in an empty 3*3 matrix. player 0 counters with a 0 in a vacant position, and play continues in turn until the 3*3 matrix is completed with five 1's and four 0's. player 0 wins if the determinant is 0 and player 1 wins otherwise. assuming both players pursue optimal strategies, who will win and how?
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 楼主| 发表于 2002-12-9 18:35 | 显示全部楼层
i've got the answer for question 3: player 0 wins. after player 1 places the the first 1, player 0 couters with a 0 at a position that is neither in the same row nor in the same column of the 1. then player 0 can make the following 0's symmetric with the central row or the central column.
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发表于 2002-12-15 20:19 | 显示全部楼层
I wish I cound attend. But the thing is that I'm so busy with the my courses this semester. How is it? Difficult? We might be able to discuss about it sometime. Or we can join the next Putnam together. Way of contact jaspermcgill@yahoo.com
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