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Quadratic polynomials and a set of integers

Source: Central American Olympiad 2007, Problem 3

June 12, 2007
quadraticsalgebrapolynomialalgebra proposed

Problem Statement

Let SS be a finite set of integers. Suppose that for every two different elements of SS, pp and qq, there exist not necessarily distinct integers a0a \neq 0, bb, cc belonging to SS, such that pp and qq are the roots of the polynomial ax2+bx+cax^{2}+bx+c. Determine the maximum number of elements that SS can have.