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Find all integers $n\ge 3$ with the following property: for all real numbers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots, b_n $ satisfying $\left\vert a_k\right\rvert+\left\lvert b_k\right\rvert = 1$ for $1\le k\le n$, there exist $x_1,x_2,\dots,x_n$ each of which is either $-1$ or $1$, such that

$$ \Big\lvert \sum _{k=1}^nx_ka_k \Big\rvert + \Big\lvert \sum _{k=1}^nx_kb_k\Big\rvert \le 1 $$

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This problem is one of IMO 2016 Shortlist problems. You can find many different solutions of it here: https://artofproblemsolving.com/community/c6h1480716p8639312

The first of many solutions is shown below.

enter image description here

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