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n quadratic residue implies an^2+bn+c quadratic residue

Source: Romania TST 5 2012, Problem 1

May 17, 2012
quadraticsalgebrapolynomialnumber theory proposednumber theory

Problem Statement

Find all triples (a,b,c)(a,b,c) of positive integers with the following property: for every prime pp, if nn is a quadratic residue mod  p\mod p, then an2+bn+can^2+bn+c is a quadratic residue mod  p\mod p.
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