List 3. Regardless of the value of the real number $k$, the line $y=2 k x$ $+k^{2}$ is tangent to the parabola $y=a x^{2}+b x+c$. Find the equation of the parabola.
Given $y=2 k x+k^{2}$ and $y=a x^{2}+b x+c$ are combined, we get
$$
a x^{2}+(b-2 k) x+\left(c-k^{2}\right)=0 .
$$

Since the line is tangent to the parabola,
thus $\Delta=(b-2 k)^{2}-4 a\left(c-k^{2}\right)=0$, which means $4(a+1) k^{2}-4 b k+b^{2}-4 a c=0$.
$\because k$ is any real number,
$\therefore$ the above equation is an identity in $k$, meaning it holds for any value of $k$.
Therefore, $\left\{\begin{aligned} 4(a+1) & =0 \\ 4 b & =0 \\ b^{2}-4 a c & =0 .\end{aligned}\right.$

Solving these, we get $a=-1, b=0, c=0$.
Thus, the equation of the parabola is $y=-x^{2}$.