LIV OM - II - Task 3

Given is the polynomial $ W(x) = x^4 - 3x^3 + 5x^2 - 9x $. Determine all pairs of different integers $ a $, $ b $ satisfying the equation
Notice that $ W(x) = (x - 1)(x - 2)(x^2 + 3) - 6 $. For $ n > 3 $, the following inequalities therefore hold






and moreover






From this, we obtain






Thus, the values of the polynomial $ W $ at the points $ -2, \pm 3, \pm 4, \ldots $ are pairwise distinct. We directly calculate that






and from inequality (1), we conclude that the values of the polynomial $ W $ at the remaining integer points are greater than $ W(3) = 18 $. Therefore, we obtain a total of four pairs of numbers satisfying the conditions of the problem:










(Note: The specific inequalities and calculations are not provided in the original text, so they are left as blank spaces in the translation.)