Problem 1. On the parabola $y=x^{2}$, four points $A, B, C, D$ are chosen such that the segments $A B$ and $C D$ intersect on the y-axis. Find the abscissa of point $D$, if the abscissas of points $A, B$ and $C$ are equal to $a, b$ and $c$ respectively.
Solution: Let the equation of the line $A B$ be $y=k x+l$, and the equation of the line $C D$ be $y=m x+l$ (the free terms are equal, since both lines intersect the ordinate axis at the same point). Then $a$ and $b$ are the roots of the equation $x^{2}=k x+l$ and from Vieta's formulas $a b=-l$. Similarly, $c d=-l$, where $d$ is the abscissa of point $D$. Hence $d=\frac{a b}{c}$.