Question 29
For a positive integer $n$, denote by $c_n$ the number of triples $(x, y, z)$ of integers such that $0 \le x \le y \le z \le x+y$ and $x+y+z=n$. Prove that, for $n \ge 2$,
$$
n \cdot c_n \le 9 \cdot (c_0 + c_1 + \dots + c_{n-2}).
$$
$$
n \cdot c_n \le 9 \cdot (c_0 + c_1 + \dots + c_{n-2}).
$$