CN-TST 2022 Problem 23

Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$ . Prove that there exist integer $p$ and $q$ , with $0 \le q \le n-1$ , such that $$\sum_{i=1}^q x_{p+2i-1} \le 1 \text{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1,$$ where the indices are take modulo $2n$ .

Note: If $q=0$ , then $\sum_{i=1}^q x_{p+2i-1}=0$ ; if $q=n-1$ , then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$ .