Question 175
Let $n$ be a positive integer. Let $x_1, x_2, \dots, x_{2n}$ be $2n$ nonnegative real numbers such that $x_1 + x_2 + \dots + x_{2n} = 4$. Prove that there exist nonnegative integers $p, q$ such that $q \le n - 1$, and that
$$
\sum_{i=1}^{q} x_{p+2i-1} \le 1, \quad \sum_{i=q+1}^{n-1} x_{p+2i} \le 1.
$$
*Remark 1: the subscripts are understood modulo $2n$, i.e. if $k \equiv \ell \pmod{2n}$, then $x_k = x_\ell$.*
*Remark 2: If $q = 0$, then the first sum is considered as 0; if $q = n - 1$, then the second sum is considered as 0.*
$$
\sum_{i=1}^{q} x_{p+2i-1} \le 1, \quad \sum_{i=q+1}^{n-1} x_{p+2i} \le 1.
$$
*Remark 1: the subscripts are understood modulo $2n$, i.e. if $k \equiv \ell \pmod{2n}$, then $x_k = x_\ell$.*
*Remark 2: If $q = 0$, then the first sum is considered as 0; if $q = n - 1$, then the second sum is considered as 0.*