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Second degree f

Source: RMO 2004, 9th grade, problem 2

March 6, 2005
functionalgebrapolynomiallogarithmscalculusintegrationinequalities

Problem Statement

Let P(n)P(n) be the number of functions f:R→Rf: \mathbb{R} \to \mathbb{R}, f(x)=ax2+bx+cf(x)=a x^2 + b x + c, with a,b,c∈{1,2,…,n}a,b,c \in \{1,2,\ldots,n\} and that have the property that f(x)=0f(x)=0 has only integer solutions. Prove that n<P(n)<n2n<P(n)<n^2, for all n≥4n \geq 4. Laurentiu Panaitopol
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