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True for nine and up (Posted on 2017-03-24) Difficulty: 3 of 5
If a set N9 = {1, 2, 3, 4, 5, 6, 7, 8, 9} of 9 numbers is split into two subsets, then at least one of them contains three terms in arithmetic progression.
The statement is not true for a set N8 of 8 integers.

Seems obvious?

Prove it.

No Solution Yet Submitted by Ady TZIDON    
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Solution By hand | Comment 2 of 3 |
The N9 partition must have a subset with at least 5 numbers.  If I didn't make any mistakes, there are four possibilities where this subset has no APs.  In each case the other subset has an AP in its 4 numbers instead.

12489 / 3567
12679 / 3458
12689 / 3457
13489 / 2567

The N8 partition is easy.  Here are the ways I found
1256 / 3478
1368 / 2457
1458 / 2367

Edited on March 25, 2017, 9:30 am
  Posted by Jer on 2017-03-24 12:39:01

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