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S (x^ 3) - S^2(x) <1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)

Source: Ukraine TST 2011 p11

May 7, 2020
PolynomialspolynomialalgebrainequalitiesInteger Polynomial

Problem Statement

Let P(x) P (x) and Q(x) Q (x) be polynomials with real coefficients such that P(0)>0 P (0)> 0 and all coefficients of the polynomial S(x)=P(x)⋅Q(x) S (x) = P (x) \cdot Q (x) are integers. Prove that for any positive x x the inequality holds: S(x2)−S2(x)≤14(P2(x3)+Q(x3)).S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})).
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