1. The number of integer pairs $(k, b)$ that make the parabola $y=x^{2}$ and the line $y=k x+b$ always have one intersection point is ( ).
(A) 1
(B) 2
(C) 3
(D) Infinite
-1.D.
Substitute $y=x^{2}$ into $y=k x+b$, and rearrange to get
$$
x^{2}-k x-b=0 \text {. }
$$

According to the condition given in the problem, equation (1) should have a unique solution, thus
$$
\Delta=k^{2}+4 b=0 \text {, }
$$

which means $k^{2}=-4 b$.
There are infinitely many integer pairs $(k, b)$ that satisfy equation (2):
$$
(0,0),(2,-1),(-2,-1),(4,-4),(-4,-4), \cdots
$$