Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is $M$ and the minimum possible number of resulting regions is $m$. What is $M^{2}+m^{2}$ ?
When the 5th chord is added, it is possible that it creates only 1 new region, which means that $m=9+1=10$

Since there are already 4 chords, then the maximum possible number of chords that the 5th chord intersects is 4 .

If the 5th chords intersects 4 chords, then it passes through 5 regions (one before the first intersection and one after each intersection) and it splits each of the 5 regions into 2 regions, which creates 5 new regions.

Since the 5th chord cannot intersect more than 4 chords, it cannot pass through more than 5 regions.

Diagrams that show each of these cases are shown below:
![](https://cdn.mathpix.com/cropped/2024_04_30_dfe99ade8da3ee5d11fdg-08.jpg?height=252&width=508&top_left_y=256&top_left_x=863)

Therefore, $M=9+5=14$ and so $M^{2}+m^{2}=14^{2}+10^{2}=296$.

ANSWER: 296