MathDB

In an island shaped like a regular polygon of

n

sides, there are airports at each vertex of the island. The island would like to add

k

new airports into the interior of the island, but it must follow the following rules:\\

1

. It must be in the interior of the island (none on borders).\\

2

. No two airports can be at the exact same location.\\

3

. Every triple of

1

new and

2

old airports must form an isoceles triangle.\\

4

. No three airports can be collinear.\\
Find the maximum value of

k

for each

n

[hide=Harder Version]Replace

1

new and

2

old with

1

old and

2

new.

Problem 4