Question 154
Given integer $n \ge 2$ and real numbers $x_1, x_2, \dots, x_n$ in the interval $[0, 1]$, prove that there exist real numbers $a_0, a_1, \dots, a_n$ satisfying simultaneously the following conditions:
$$
(1)
\begin{aligned}
a_0 &+ a_n = 0; \\
a_i &\le 1, \text{ for every } i = 0, 1, \dots, n;
\end{aligned}
$$
$$
(2) \quad |a_i| \le 1, \text{ for every } i = 0, 1, \dots, n;
$$
$$
(3) \quad |a_i - a_{i-1}| = x_i, \text{ for every } i = 1, 2, \dots, n.
$$
$$
(1)
\begin{aligned}
a_0 &+ a_n = 0; \\
a_i &\le 1, \text{ for every } i = 0, 1, \dots, n;
\end{aligned}
$$
$$
(2) \quad |a_i| \le 1, \text{ for every } i = 0, 1, \dots, n;
$$
$$
(3) \quad |a_i - a_{i-1}| = x_i, \text{ for every } i = 1, 2, \dots, n.
$$