Problem 6

Prove that there exists a convex 1990-gon whose angles are all equal and whose side lengths are

1^2,2^2,\ldots,1990^2

in some order.

Step 1 of 4: Encode the equiangular sides as complex vectors

Previous step Next step

\omega=e^{2\pi i/1990},\qquad v_{i,j,k}=r(i,j,k)^2(-1)^ia^jb^k

Detailed analysis

Let $\omega$ be a primitive 1990th root of unity. We will assign the length $r(i,j,k)^2$ to the direction $(-1)^ia^jb^k$ . These directions are precisely the 1990 successive directions whose arguments increase by $2\pi/1990$ . If the vector sum is zero, the corresponding positive side vectors close to form an equiangular polygon; the constant positive turn makes it convex.