1

Part of 1983 Federal Competition For Advanced Students, P2

Problems(1)

infinitely many values

Source: Austria 1983

For every natural number

x

Q(x)

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))\plus{}P(Q(x))\plus{}Q(P(x))\plus{}P(P(x))\equal{}n.

algebra unsolvedalgebra