ÖMO 1983 Problem 1

For every natural number $x$ , let $Q(x)$ be the sum and $P(x)$ the product of the (decimal) digits of $x$ . Show that for each $n \in \mathbb{N}$ there exist infinitely many values of $x$ such that:

$Q(Q(x))+P(Q(x))+Q(P(x))+P(P(x))=n$ .