49th Mathematical Olympiad in Ukraine

Let the equation of the line $l_1$ be $y=kx+d$. Then abscissas of points $A$ and $B$ are defined using the equality $a{x}^2+bx+c=kx+d$. Therefore these abscissas $x_1,x_2$ satisfy that quadratic equation, then according to Vieta's formula we have equality $x_1+x_2=\frac{-b+k}{a}$.

The abscissa of point $C$ is defined using the same equality

$a{x}^2+bx+c=kx+d$

on condition that the line $l_2$ is tangent to this parabola. Then the abscissa of point $C$ is $\frac{k-b}{2a}=\frac{x_1+x_2}{2}$, which completes the proof.