cannot be satisfied simultaneously.
Suppose, without loss of generality, a < b < c.
By the polynomial remainder theorem, P(c) − P(a) = (c − a)Q(c), for some polynomial Q, with integer coefficients.
Hence c − a divides a − b.
But this is impossible, as |c − a| > |a − b|, and a b.
Therefore the above system of equations cannot be satisfied simultaneously.