Let f∈ Q[x] be a polynomial without multiple roots and deg f≥ 2. We give conditions for f= x2+ bx+ c under which the Diophantine equation 2 f(x) f(y) = f(z) (f(x) + f(y)) has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for f= x2+ bx with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials...
