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China Team Selection Test 2017 TST 2 Day 2 Q5

Source: China Shanghai ,Mar 12, 2017

March 12, 2017
number theoryalgebrapolynomialChina TST

Problem Statement

Let φ(x) \varphi(x) be a cubic polynomial with integer coefficients. Given that φ(x) \varphi(x) has have 3 distinct real roots u,v,wu,v,w and u,v,wu,v,w are not rational number. there are integers a,b,c a, b,c such that u=av2+bv+cu=av^2+bv+c. Prove that b2−2b−4ac−7b^2 -2b -4ac - 7 is a square number .
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