2012 IMO Shortlist C7

There are given $2^{500}$ points on a circle labeled $1,2, \ldots, 2^{500}$ in some order. Prove that one can choose 100 pairwise disjoint chords joining some of these points so that the 100 sums of the pairs of numbers at the endpoints of the chosen chords are equal.

圆上有给定的 $2^{500}$ 点，按某种顺序标记为 $1,2, \ldots, 2^{500}$。证明可以选择 100 个成对的不相交弦来连接其中一些点，以便所选弦端点处的数字对的 100 个和相等。