ISL 2005 N2 (BOMC 2)

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sourav das
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ISL 2005 N2 (BOMC 2)

Unread post by sourav das » Mon Apr 02, 2012 7:23 pm

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.

Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.

Really a good one :)
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zadid xcalibured
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Re: ISL 2005 N2 (BOMC 2)

Unread post by zadid xcalibured » Mon Apr 02, 2012 7:44 pm

are the terms $a_1,a_2,....$in increasing or decreasing order?
Last edited by zadid xcalibured on Mon Apr 02, 2012 7:50 pm, edited 1 time in total.

sourav das
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Re: ISL 2005 N2 (BOMC 2)

Unread post by sourav das » Mon Apr 02, 2012 7:47 pm

Nothing else is given. So not necessarily increasing or decreasing.
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zadid xcalibured
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Re: ISL 2005 N2 (BOMC 2)

Unread post by zadid xcalibured » Mon Apr 02, 2012 7:49 pm

they appear at most once of course. :mrgreen:

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Re: ISL 2005 N2 (BOMC 2)

Unread post by sourav das » Mon Apr 02, 2012 7:56 pm

zadid xcalibured wrote:they appear at most once of course. :mrgreen:
I knew that this problem might occur (I also felt the same at first). But the question wants you to prove that all integers(all in $Z$) occurs in the sequence and they occurs exactly once.
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When you go down, when you go down down......
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*Mahi*
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Re: ISL 2005 N2 (BOMC 2)

Unread post by *Mahi* » Mon Apr 02, 2012 8:17 pm

One of the cooooolest NT problems of IMO till date...awesome... :mrgreen:
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Phlembac Adib Hasan
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Re: ISL 2005 N2 (BOMC 2)

Unread post by Phlembac Adib Hasan » Tue Apr 03, 2012 12:48 pm

*Mahi* vaia wrote:One of the cooooolest NT problems of IMO till date...awesome... :mrgreen:
আমিও একমত।অনেকদিন এত মজার NT সলভ করি নাই। :D
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