There are given
![2^{500}](/media/m/0/7/3/07355210f35d5ad0d8d681a0d4aeb053.png)
points on a circle labeled
![1,2,\ldots ,2^{500}](/media/m/c/4/2/c42cb4723680025fd24d1290bd910be4.png)
in some order. Prove that one can choose
![100](/media/m/c/c/c/ccc0563efabf7c1a3d81b0dc63f5b627.png)
pairwise disjoint chords joining some of theses points so that the
![100](/media/m/c/c/c/ccc0563efabf7c1a3d81b0dc63f5b627.png)
sums of the pairs of numbers at the endpoints of the chosen chord are equal.
%V0
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 theses points so that the $100$ sums of the pairs of numbers at the endpoints of the chosen chord are equal.