2012-12 Big partial sum

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that $\bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.$

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2012-2 sum with a permutation

Let n be a positive integer and let Sn be the set of all permutations on {1,2,…,n}. Assume $$x_1+x_2 +\cdots +x_n =0$$ and $$\sum_{i\in A} x_i\neq 0$$ for all nonempty proper subsets A of {1,2,…,n}. Find all possible values of$\sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}.$

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Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist $$y_1,y_2,\ldots,y_k$$ with $$k\le 53$$ such that $$x=\sum_{i=1}^k y_i^4$$.