Question 3
Let $x_1$, $x_2$, …, $x_n$ be real numbers satisfying the conditions: $$
\begin{aligned}|x_1 + x_2 + \dots + x_n| &= 1 \\
|x_i| &\le \frac{n+1}{2} \qquad \text{for } i= 1,2, \dots, n\end{aligned}
$$ Show that there exists a permutation $y_1$, $y_2$, …, $y_n$ of $x_1$, $x_2$, …, $x_n$ such that $$
| y_1 + 2 y_2 + \dotsb + n y_n | \leq \frac {n + 1}{2}.
$$
\begin{aligned}|x_1 + x_2 + \dots + x_n| &= 1 \\
|x_i| &\le \frac{n+1}{2} \qquad \text{for } i= 1,2, \dots, n\end{aligned}
$$ Show that there exists a permutation $y_1$, $y_2$, …, $y_n$ of $x_1$, $x_2$, …, $x_n$ such that $$
| y_1 + 2 y_2 + \dotsb + n y_n | \leq \frac {n + 1}{2}.
$$