2016 Set Operations Problem

Since any 2016 elements are removed, suppose we remove the integers from 1 to 2016. Then the smallest possible sum of 2016 of the remaining elements is $2017+2018+\cdots + 4032 = 1008 \cdot 6049 = 6097392$ so clearly $N\ge 6097392$ . We will show that $N=6097392$ works.

$1,2\cdots 6097392$ contains the integers from 1 to 6048, so pair these numbers as follows:

$1, 6048$

$2, 6047$

$3, 6046$

$\cdots$

$3024, 3025$

When we remove any 2016 integers from the set $1,2\cdots N$ , clearly we can remove numbers from at most 2016 of the 3024 pairs above, leaving at least 1008 complete pairs. To get a sum of N, simply take these 1008 pairs, all of which sum to 6049. The sum of these 2016 elements is $1008 \cdot 6049 = 6097392$ , as desired.

We have shown that N must be at least 6097392, and that this value is attainable. Therefore our answer is 6097392.