CGMO 2022 Problem 1

Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements: (1) $x_0=0$ ; (2)For any integer $i,1\leq i\leq 100$ ,we have $1\leq x_i-x_{i-1}\leq 2$ . Find the greatest positive integer $k\leq 100$ ,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have $$x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.$$