## Task B-2.3.

The equation $y=x^{2}-2(m+1) x+2 m^{2}+2018, m \in \mathbb{R}$ defines a set of parabolas. To which set of points do the vertices of all such parabolas belong? Determine the sum of the coordinates of the point from that set which is closest to the $x$-axis.
## Solution.

The coordinates of the vertex are of the form:

$$
\begin{aligned}
& x_{0}=m+1 \\
& y_{0}=m^{2}-2 m+2017
\end{aligned}
$$

If we express $m=x_{0}-1$ from the first equation and substitute it into the second to obtain the relationship between the coordinates of the vertex, we get

$$
y_{0}=\left(x_{0}-1\right)^{2}-2\left(x_{0}-1\right)+2017=x_{0}^{2}-4 x_{0}+2020
$$

or $y_{0}=\left(x_{0}-2\right)^{2}+2016$.

We conclude that the vertices of all parabolas in the given set lie on the parabola $y=(x-2)^{2}+2016$.

The point on this parabola that is closest to the $x$-axis is its vertex, i.e., $T(2,2016)$. Therefore, the desired sum of the coordinates is 2018.