Question 6
(CAN 2) Suppose that $\{x_1,x_2,\ldots,x_n\}$ are positive integers for which $x_1+x_2+\cdots+x_n=2(n+1)$. Show that there exists an integer $r$ with $0\le r\le n-1$ for which the following $n-1$ inequalities hold:
$$\begin{aligned}x_{r+1}+\cdots+x_{r+i}&\le2i+1&&\forall i,\ 1\le i\le n-r;\\x_{r+1}+\cdots+x_n+x_1+\cdots+x_i&\le2(n-r+i)+1&&\forall i,\ 1\le i\le r-1.\end{aligned}$$
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r$.
$$\begin{aligned}x_{r+1}+\cdots+x_{r+i}&\le2i+1&&\forall i,\ 1\le i\le n-r;\\x_{r+1}+\cdots+x_n+x_1+\cdots+x_i&\le2(n-r+i)+1&&\forall i,\ 1\le i\le r-1.\end{aligned}$$
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r$.