What is the equation of the parabola that touches the lines given by the equations $y=x+2$, $y=3x-2$, and $2x+y+7=0$, and whose axis is parallel to the $y$-axis?
The axis of the parabola is parallel to the $y$-axis, so its equation is of the form $y=a x^{2}+b x+c$. Since it is tangent to the lines given by the equations $y=x+2$, $y=3 x-2$, and $y=-2 x-7$, the equations

\[
\begin{aligned}
& a x^{2}+b x+c=x+2 \\
& a x^{2}+b x+c=3 x-2 \\
& a x^{2}+b x+c=-2 x-7
\end{aligned}
\]

each have exactly one solution for $x$. This condition implies that the discriminants of the quadratic equations (1), (2), and (3) must be zero, i.e.,

\[
\begin{aligned}
& (b-1)^{2}-4 a(c-2)=0 \\
& (b-3)^{2}-4 a(c+2)=0 \\
& (b+2)^{2}-4 a(c+7)=0
\end{aligned}
\]

From the system of equations (4), (5), and (6), $a$, $b$, and $c$ can be determined: the difference between equations (6) and (5), and between equations (6) and (4) is

\[
\begin{array}{r}
10 b-5-20 a=0 \\
6 b+3-36 a=0
\end{array}
\]

From this system of equations, $a=\frac{1}{4}, b=1$. With these values, any of the equations yields $c=2$. The obtained values of $a$, $b$, and $c$ satisfy equations (4), (5), and (6), so the parabola with the equation $y=\frac{1}{4} x^{2}+x+2$ is the only one that meets the conditions of the problem.

Imreh Csanád (Szeged, Ságvári E. Gyak. Gimn., III. o. t.)