Question 48
Pablo was copying the following problem:
Consider all sequences of $2004$ real numbers $(x_0, x_1, x_2, ..., x_{2001})$, such that
$$
\begin{array}{l}
x_0 = 1, \\
0 \le x_1 \le 2x_0, \\
0 \le x_2 \le 2x_1, \\
\vdots \\
0 \le x_{2003} \le 2x_{2002}.
\end{array}
$$
Among all these sequences, determine the one for which the following expression takes its largest value: $S = ...$.
When Pablo was going to copy the expression from $S$, they erased his blackboard. The only thing he could remember was that $S$ was of the form
$$
S = \pm x_1 \pm x_2 \pm \cdots \pm x_{2001} \pm x_{2002}
$$
where the last term, $x_{2001}$, had coefficient $+1$, and the previous ones had coefficient $+1$ or $-1$. Show that Paul, despite not having the complete statement, can determine with certainty the solution to the problem.
Consider all sequences of $2004$ real numbers $(x_0, x_1, x_2, ..., x_{2001})$, such that
$$
\begin{array}{l}
x_0 = 1, \\
0 \le x_1 \le 2x_0, \\
0 \le x_2 \le 2x_1, \\
\vdots \\
0 \le x_{2003} \le 2x_{2002}.
\end{array}
$$
Among all these sequences, determine the one for which the following expression takes its largest value: $S = ...$.
When Pablo was going to copy the expression from $S$, they erased his blackboard. The only thing he could remember was that $S$ was of the form
$$
S = \pm x_1 \pm x_2 \pm \cdots \pm x_{2001} \pm x_{2002}
$$
where the last term, $x_{2001}$, had coefficient $+1$, and the previous ones had coefficient $+1$ or $-1$. Show that Paul, despite not having the complete statement, can determine with certainty the solution to the problem.