RO-NMO 2004 Problem 92

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$ , $f(x)=a x^2 + b x + c$ , with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$ , for all $n \geq 4$ .

Laurentiu Panaitopol