OME 1969 Problem 6

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: $$P(x) \le P(x)^2; \,\, P(x) < 1 + P(x)^2; \,\, P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$ , at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$ .