BdMO Online Forum

$20$ points are taken inside a regular hexagon of side length $1$. Prove that there exists at least two points whose distance is less than or equal to $\frac12$.

I have tried to break the hexagon into several parts and apply PHP, but this seems a little tricky...

If PHP is useless in this, is there any way we can at least start to make progress? Because if there were 25 points then PHP would have worked fine. 20 points make it really difficult.

Mehrab4226 wrote: ↑

Wed Jan 20, 2021 11:33 pm

If PHP is useless in this, is there any way we can at least start to make progress? Because if there were 25 points then PHP would have worked fine. 20 points make it really difficult.

I can divide the hexagon into 18 pieces nicely, but those pieces aren't small enough

Anindya Biswas wrote: ↑

Wed Jan 20, 2021 11:48 pm

I can divide the hexagon into 18 pieces nicely, but those pieces aren't small enough

SAME!

21 points also work easily...(Though it can be divided into several hexagons to meet the condition Hint: If the side length of regular hex is a then the biggest corner is 2a)

Asif Hossain wrote: ↑

Tue Feb 09, 2021 8:53 pm

21 points also work easily...(Though it can be divided into several hexagons to meet the condition Hint: If the side length of regular hex is a then the biggest corner is 2a)

Yeah I actually figured this 20 points problem out some days after this post, now kept that solution as a secret for others to explore.

Actually i asked you that question on facebook and i inherited that ingenious solution from you

Asif Hossain wrote: ↑

Wed Feb 10, 2021 9:39 pm

Actually i asked you that question on facebook and i inherited that ingenious solution from you

Wow