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Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$ and $x_1\leq x_2 \leq\cdots\leq x_n$ and $y_1\leq y_2 \leq\cdots\leq y_n$ and

$$1-\frac{1}{n} \lt x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \lt 1 .$$

Can we prove that $$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$

for all permutations $\sigma$?

Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$

$$ x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \gt 1-\frac{1}{n} .$$

Can we prove that $$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$

for all permutations $\sigma$?

Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$ and

$$x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \gt 1-\frac{1}{n}.$$

Can we prove that $$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$

for all permutations $\sigma$?

Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$ and $x_1\leq x_2 \leq\cdots\leq x_n$ and $y_1\leq y_2 \leq\cdots\leq y_n$ and

$$1-\frac{1}{n} \lt x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \lt 1 .$$

Can we prove that $$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$

for all permutations $\sigma$?