An average problem

Discrete Mathematics Level 5

The natural numbers from 1 to 2015 are placed on a line in any order, so that no two numbers lie on the same point and that their average mean does not lie between those two numbers.

For example: You place the 1 at the very left, 2015 at the very right. This leads to a problem because $$(1+2015) \over 2$$ = 1008 is their average and has to lie between 1 and 2015., therefore those two numbers would both not suffice the property.

However one could place numbers in the following order 2015 1 2 3 ..... 2013 2014. Now 2015 would suffice the property stated above. However the numbers 1 and 3 now make a problem since the number 2 lies between them..

Is it possible to find an arrangement of those numbers, so that all numbers suffice the property given above? If not, what is the maximum amount of numbers to hold this property?

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