1. Can the numbers from 1 to 2016 be arranged in a circle so that all the sums of adjacent numbers are different?
Solution. Consider the arrangement $1,2,3, \ldots, 2014,2016,2015$. Let's write down the sums of adjacent numbers: $1+2=3,2+3=5,3+4=7, \ldots, 2013+2014=4027,2014+$ $2016=4030,2016+2015=4029,2015+1=2016$. In this list, the odd numbers $3,5, \ldots, 4027,4029$ and even numbers $4030,2016$ will each appear once.

Answer. Yes, it is possible.