Here is a fun theorem: Consider 31 points evenly spaced on a circle, and color each of them arbitrarily blue or red. Then we can always find 5 points with the same color that divide the circle into arcs proportional to 1 : 2 : 4 : 8 : 16. The arcs need not be in the order suggested by the proportion. That is, 1 : 4 : 8 : 2 : 16 counts as a success!
One of my favorite open problem is whether the generalization of this holds:
Stromquist's conjecture: For any k>= 3, consider (2^k)-1 points evenly spaced on a circle, and color each of them arbitrarily blue or red. Then we can always find k points with the same color that divide the circle into arcs proportional to 1:2:4: ... :2^(k-1), but not necessarily in this order.
I managed to prove this up to k=7 using a computer. Can you push it further?
