IMO 1989 LL ROM93

For 
hi : N oZ let us define M
hi = {f : N oZ; f(x) >

imolonglistmathematicsolympiad

IMO 1989 LL ROM93

Origin: ROM

Problem

For \Phi : N \toZ let us define M\Phi = {f : N \toZ; f(x) > F(\Phi(x)), \forallx \inN}. (a) Prove that if M\Phi1 = M\Phi2 ̸= \emptyset, then \Phi1 = \Phi2. (b) Does this property remain true if M\Phi = {f : N \toN; f(x) > F(\Phi(x)), \forallx \inN}?