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6 replies to this topic

### #1 bushindo

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Posted 02 January 2010 - 03:47 AM

I saw this problem in a test long ago.

Assume that we have a sequence of 17 distinct integers. Show that there exist at least 1 subsequence of 5 integers that form an increasing or decreasing sequence.

For instance, let the sequence of 17 distinct integers be
```9, 13,  6,  1, 11,  7, 16,  3, 15, 10,  2,  5, 12,  4, 14,  8, 17
```

Then there is a increasing subsequence of length 5 in (1,3,5,8,17). There is also a decreasing subsequence of (16,15,10,5,4). Show that the statement in the second paragraph is true for all sequences of 17 distinct integers.
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### #2 alanklm

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Posted 02 January 2010 - 10:49 AM

Gm.
17, 5. Strange dependence...

bushindo, Is solution simple enough?
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### #3 magician

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Posted 02 January 2010 - 04:33 PM

I think I have a solution, but if everyone else is like me, a spoiler button is like a big sign saying "click here", so I'll hold off for a bit and just give a hint.

Spoiler for something to note

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### #4 superprismatic

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Posted 03 January 2010 - 03:14 AM

Spoiler for as I recall

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### #5 bushindo

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Posted 03 January 2010 - 06:36 PM

Spoiler for as I recall

That's some childhood you got, maybe it's a polish thing. I got mostly spanks on my mother's knee. I'll supplement the hint that magician gave earlier

Spoiler for

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### #6 superprismatic

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Posted 04 January 2010 - 10:55 PM

Well, it seems nobody wants to provide a proof so
Spoiler for here's mine

Edited by superprismatic, 04 January 2010 - 10:58 PM.

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### #7 bushindo

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Posted 05 January 2010 - 12:12 AM

Well, it seems nobody wants to provide a proof so

Spoiler for here's mine

That's a nice proof. I didn't think of that. Here's a clumsier proof using induction
Spoiler for

Edited by bushindo, 05 January 2010 - 12:17 AM.

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