IMO 1985 LL ROM70

Let C be a class of functions f : N oN that contains the

imolonglistmathematicsolympiad

IMO 1985 LL ROM70

Origin: ROM

Problem

Let C be a class of functions f : N \toN that contains the functions S(x) = x + 1 and E(x) = x −[\sqrtx]2 for every x \inN. ([x] is the integer part of x.) If C has the property that for every f, g \inC, f + g, fg, f ◦g \inC, show that the function max(f(x) −g(x), 0) is in C.