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P(x) <= P(x)^2; P(x) < 1 + P(x)^2; P(x) <=1/2 +1/2 P(x)^2

Source: Spanish Mathematical Olympiad 1969 P6

December 4, 2022
polynomialalgebrainequalities

Problem Statement

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: P(x)≤P(x)2;   P(x)<1+P(x)2;   P(x)≤12+12P(x)2.P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2. Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials P(x)P(x) and Q(x)Q(x) , find another M(x)M(x) such that for every value of xx, at the same time −M(x)<P(x)<M(x)-M(x) < P(x)<M(x) and −M(x)<Q(x)<M(x)-M(x)< Q(x)<M(x).
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