This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions)

Let $A\subseteq X=\{1,2,3,\cdots,n\}$, $B=X\setminus A$. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such that: $$\begin{cases} \forall a\in A,|x^a_{1}+x^a_{2}+\cdots+x^a_{m}|\le\frac{1}{a} \text{ for all } m \le k\\ \forall b\in B,|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1. \end{cases}$$



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