NZ-NZMO 2025 Problem 6

Determine the largest real number $M$ such that for each infinite sequence $x_{0}, x_{1}, x_{2}, \ldots$ of real numbers satisfying $x_{0} = 1$ , $x_{1} = 3$ and

$$x_{0} + x_{1} + \dots +x_{n - 1}\geq 3x_{n} - x_{n + 1} \mathrm{for~all~}n\geq 1,$$

the inequality

$$\frac{x_{n + 1}}{x_{n}} >M$$

holds for all $n \geq 0$ .