IMO 1987 LL LUX39

Origin: LUX

Problem

Let A be a set of polynomials with real coefficients and let them satisfy the following conditions: (i) if f \inA and deg f \leq1, then f(x) = x −1; (ii) if f \inA and deg f \geq2, then either there exists g \inA such that f(x) = x2+deg g + xg(x) −1 or there exist g, h \inA such that f(x) = x1+deg gg(x) + h(x); (iii) for every f, g \inA, both x2+deg f + xf(x) −1 and x1+deg ff(x) + g(x) belong to A. Let Rn(f) be the remainder of the Euclidean division of the polynomial f(x) by xn. Prove that for all f \inA and for all natural numbers n \geq1 we have Rn(f)(1) \leq0 and Rn(f)(1) = 0 ⇒Rn(f) \inA.