Question 100
Let $n \geqslant 2$ be an integer and $C \gt 0$ a real constant. Suppose that there exists a sequence $x_{1}, x_{2}, \ldots, x_{n}$ of real numbers, not all zero, such that:
- $x_{1}+x_{2}+\ldots+x_{n}=0$;
- for every index $1 \leqslant i \leqslant n$, we have either $x_{i} \leqslant x_{i+1}$ or $x_{i} \leqslant x_{i+1}+C x_{i+2}$ (where we define $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$).
1. Show that necessarily $C \geqslant 2$.
2. Suppose that $C=2$; show that $n$ is necessarily even.
- $x_{1}+x_{2}+\ldots+x_{n}=0$;
- for every index $1 \leqslant i \leqslant n$, we have either $x_{i} \leqslant x_{i+1}$ or $x_{i} \leqslant x_{i+1}+C x_{i+2}$ (where we define $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$).
1. Show that necessarily $C \geqslant 2$.
2. Suppose that $C=2$; show that $n$ is necessarily even.