This is inspired by Mehul Chaturvedi's question Equally divided 10 points.

Consider all sets of 10 points in the plane, in which no 3 are collinear. A line \(l\) that connects 2 of these points is a **dividing line** if it divides the remaining points into 2 equal regions of 4 points each. Over all configurations, what is the maximum number of dividing lines?

By a simple counting argument, we can show that there are at least 5 dividing lines in any configuration.

If the 10 points form a convex set (ie no points are in the interior of the convex hull), then we can prove that there are exactly 5 dividing lines.

To get more than 5, consider the vertices of a 9-gon, along with the center. Then, any vertex that is connected to the center gives us a dividing line, and so there are (at least) 9 such lines.

Can we do better?

Can we generalize this to \( 10 = 2n \)?

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.