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Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that $$ \sum\limits_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum\limits_{j=1}^n a_j\cdot x_{\pi{(j)}} >0 $$ for any permutation $\pi$ of $\{1,2,\ldots,n\}$ different from identity?

Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that $$ \sum_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum_{j=1}^n a_j\cdot x_{\pi(j)} >0 $$ for any permutation $\pi$ of $\{1,2,\ldots,n\}$ different from identity?

Let $x_1< x_2< \cdots< x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j \neq 0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that $$ \sum\limits_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum\limits_{j=1}^n a_j\cdot x_{\pi{(j)}} >0 $$ for all permutation $\pi$ of $\{1,2,\ldots,n\}$ different from identity?

Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that $$ \sum\limits_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum\limits_{j=1}^n a_j\cdot x_{\pi{(j)}} >0 $$ for any permutation $\pi$ of $\{1,2,\ldots,n\}$ different from identity?

Let $x_1< x_2< ...< x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j \neq 0$. Do there always exist $n$ integers $a_1,a_2,...,a_n$ such that $$ \sum\limits_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum\limits_{j=1}^n a_j\cdot x_{\pi{(j)}} >0 $$ for all permutation $\pi$ of $\{1,2,...,n\}$ different from identity?

Let $x_1< x_2< \cdots< x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j \neq 0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that $$ \sum\limits_{j=1}^n a_j\cdot x_j <0 \quad\text{and such that}\quad \sum\limits_{j=1}^n a_j\cdot x_{\pi{(j)}} >0 $$ for all permutation $\pi$ of $\{1,2,\ldots,n\}$ different from identity?