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number of triples such that the roots of ax^2+bx+c=0 are both integers

Source: 2024 5th OMpD L3 P3 - Brazil - Olimpíada Matemáticos por Diversão

October 16, 2024
algebra

Problem Statement

For each positive integer n n , let f(n) f(n) be the number of ordered triples (a,b,c) (a, b, c) such that a,b,c∈{1,2,…,n} a, b, c \in \{1, 2, \ldots, n\} and that the two roots (possibly equal) of the quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 are both integers.
(a) Prove that for every positive real number C C , there exists a positive integer nC n_C such that for all integers n≥nC n \geq n_C , we have f(n)>C⋅n f(n) > C \cdot n .
(b) Prove that for every positive real number C C , there exists a positive integer nC n_C such that for all integers n≥nC n \geq n_C , we have f(n)<C⋅n20252024 f(n) < C \cdot n^{\frac{2025}{2024}} .
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