TSTST 2024 Problem 2

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0) = Q(0) = 1$ , there is no nonconstant polynomial dividing both $P$ and $Q$ , and

$$1 + \frac{x}{1 + \frac{2x}{1 + \frac{\cdots}{1 + (p - 1)x}}} = \frac{P(x)}{Q(x)}.$$

Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$ , and all coefficients of $Q$ are not divisible by $p$ .