We call (F, c) a good pair if the following three conditions are satisfied:
(1) F(x)=a_{0}+a_{1} x+\cdots a_{m} x^{m},(m \geq 1) is a non-constant polynomial with integer coefficients.
(2) c is a real number that is not an integer.
(3) F(c) is an integer.
For example, both \left(6 x, \frac{1}{3}\right) and \left(1+x^{3}, 5^{1 / 3}\right) are good pairs, but none of the following pairs \left(6 x, \frac{1}{4}\right), (6 x, 2),\left(\frac{x}{6}, \frac{1}{3}\right),\left(\frac{x^{2}}{6}, 6\right) is good.
a. Let c=\frac{1}{2}. Give an example of F such that (F, c) is a good pair but (F, c+1) is not.
b. Let c=\sqrt{2}. Give an example of F such that both (F, c) and (F, c+1) are good pairs.
c. Show that for any good pair (F, c), if c is rational then there exists infinitely many non-zero integers n such that (F, c+n) is also a good pair.
d. Show that if (F, c+n) is a good pair for every integer n, then c is rational.