Question 158
Consider all sequences of real numbers $x_0, x_1, x_2, \dots, x_{100}$ satisfying the following conditions:
(1) $x_0 = 0$;
(2) For any integer $i$, $1 \le i \le 100$, $1 \le x_i - x_{i-1} \le 2$ holds.
Find the largest positive integer $k \le 100$, such that
$$
x_k + x_{k+1} + \dots + x_{100} \ge x_0 + x_1 + \dots + x_{k-1}
$$
holds for every such sequence $x_0, x_1, x_2, \dots, x_{100}$.
(1) $x_0 = 0$;
(2) For any integer $i$, $1 \le i \le 100$, $1 \le x_i - x_{i-1} \le 2$ holds.
Find the largest positive integer $k \le 100$, such that
$$
x_k + x_{k+1} + \dots + x_{100} \ge x_0 + x_1 + \dots + x_{k-1}
$$
holds for every such sequence $x_0, x_1, x_2, \dots, x_{100}$.