10-7. Parabola $\Pi_{1}$ with branches directed upwards passes through points with coordinates $(10,0)$ and $(13,0)$. Parabola $\Pi_{2}$ with branches directed upwards also passes through the point with coordinates $(13,0)$. It is also known that the vertex of parabola $\Pi_{1}$ bisects the segment connecting the origin and the vertex of parabola $\Pi_{2}$. At what abscissa does parabola $\Pi_{2}$ intersect the $O x$ axis again?
Answer: 33.

Solution. We will use the following fact twice: if $x_{1}$ and $x_{2}$ are the x-coordinates of the points where the parabola intersects the x-axis, then the x-coordinate of the vertex is $\frac{x_{1}+x_{2}}{2}$ (the x-coordinate of the vertex is the midpoint of the segment with endpoints $x_{1}$ and $x_{2}$).

![](https://cdn.mathpix.com/cropped/2024_05_06_1d29d5cb430346a58eb5g-28.jpg?height=360&width=717&top_left_y=434&top_left_x=675)

Applying this fact, we get that the x-coordinate of the vertex of the parabola $\Pi_{1}$ is $\frac{10+13}{2}=11.5$, and then the x-coordinate of the vertex of the parabola $\Pi_{2}$ is $2 \cdot 11.5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1d29d5cb430346a58eb5g-28.jpg?height=386&width=734&top_left_y=972&top_left_x=661)

If we denote by $t$ the sought x-coordinate of the second point of intersection of the parabola $\Pi_{2}$ with the x-axis, then, applying the fact again, we get $\frac{13+t}{2}=2 \cdot 11.5$, from which $t=33$.

Comment. The condition that the branches of the parabolas are directed upwards was not used in the solution.