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Maximum number of rational roots in a family of quadratics

Source: INAMO 2023 P8 (OSN 2023)

August 30, 2023
algebrapolynomialquadratic equationquadraticsmaximizationIndonesia

Problem Statement

Let a,b,ca, b, c be three distinct positive integers. Define S(a,b,c)S(a, b, c) as the set of all rational roots of px2+qx+r=0px^2 + qx + r = 0 for every permutation (p,q,r)(p, q, r) of (a,b,c)(a, b, c). For example, S(1,2,3)={−1,−2,−1/2}S(1, 2, 3) = \{ -1, -2, -1/2 \} because the equation x2+3x+2x^2+3x+2 has roots −1-1 and −2-2, the equation 2x2+3x+1=02x^2+3x+1=0 has roots −1-1 and −1/2-1/2, and for all the other permutations of (1,2,3)(1, 2, 3), the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in S(a,b,c)S(a, b, c).
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