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
ISL 2005 N2 (BOMC 2)
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You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- zadid xcalibured
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Re: ISL 2005 N2 (BOMC 2)
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.
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Re: ISL 2005 N2 (BOMC 2)
Nothing else is given. So not necessarily increasing or decreasing.
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- zadid xcalibured
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Re: ISL 2005 N2 (BOMC 2)
they appear at most once of course.
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Re: ISL 2005 N2 (BOMC 2)
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.zadid xcalibured wrote:they appear at most once of course.
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: ISL 2005 N2 (BOMC 2)
One of the cooooolest NT problems of IMO till date...awesome...
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Re: ISL 2005 N2 (BOMC 2)
আমিও একমত।অনেকদিন এত মজার NT সলভ করি নাই।*Mahi* vaia wrote:One of the cooooolest NT problems of IMO till date...awesome...
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