Problem 6
Prove that there exists a convex 1990-gon whose angles are all equal and whose side lengths are in some order.
Step 4 of 4: The vector sum vanishes by three geometric sums
Detailed analysis
Put s=1+199j+k. Summing first over i gives minus 995 squared times the sum of a^j b^k, minus 1990 times the sum of s a^j b^k. The first term is zero because the sum of the fifth powers a^j is zero. In the second term, the part (1+k) is again zero after summing over j, while the part 199j is zero after summing over k. Thus the whole vector sum is zero, and the construction proves the claim.