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Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I show that there's some other set, say M such that $$\left|\sum_{j\in M} z_j\right|\ge \frac{1}{8} \sum_{k=1}^n |z_k|. $$

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Won't the same set work? Or do you need it to be distinct from $J$? –  Aryabhata Jan 25 '12 at 18:59
That's what I thought, but I don't know how to get the 1/8. –  Joel Jan 25 '12 at 19:00
Show that $8 \ge \pi$. btw, what is the source of this problem? –  Aryabhata Jan 25 '12 at 19:01
Related post: math.stackexchange.com/q/91939/13425. (@Aryabhata: Pinging you since I think you might be interested in that post.) –  Srivatsan Jan 25 '12 at 19:04
@Joel: See the related post Srivatsan referred to. I ask again: Do you know the source? Knowing that might clarify the problem a bit. (and it would help folks who want to read further). –  Aryabhata Jan 25 '12 at 20:01

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