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Putnam 1981 A5

Source: Putnam 1981

March 31, 2022
Putnampolynomialroots

Problem Statement

Let P(x)P(x) be a polynomial with real coefficients and form the polynomial Q(x)=(x2+1)P(x)P′(x)+x(P(x)2+P′(x)2).Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ). Given that the equation P(x)=0P(x) = 0 has nn distinct real roots exceeding 11, prove or disprove that the equation Q(x)=0Q(x)=0 has at least 2n−12n - 1 distinct real roots.
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