Given a sequence of complex numbers that $$\sum_{l=1}^{n}|z_l|=1$$ I want to find the following number $$\inf\left\vert\sup\limits_{I\subset \mathbb{Z}_n}\sum\limits_{i\in I}z_i\right\vert$$ This arises from the third problem of CMO, 1986.
Somebody told me that the result is $1/\pi$ which should use some complex analysis. Besides, I am also interested in the example that is sufficiently close to the infimum.
