Baltic Way 2017 · Problem 15

Answer: 0 if $n=3,4$ and $n-3$ for $n \geq 5$.

If $n=3,4$ then any such $n$-gon is a triangle, resp. a rhombus, therefore the answer is 0 .

If $n=5$ then consider a triangle with side lengths 2,2,1. Now move the vertex between sides of length 2 towards the opposite side by 0.0000001 units. Consider the triangle as a closed physical chain of links of length 1 that are aluminium tubes and through them is a closed rubber string. So deform the chain on a level surface so that links of the chain move towards the inside of the triangle, by fixing the vertices of the triangle to the surface. Geometrically, the links which are on sides of length 2 are now distinct chords of a large circle, attached to each other by their endpoints.

If $n>5$ then first consider a triangle of integer sides lengths, with sides of as equal lengths as possible, so that the sum of side lengths is $n$. Imagine a closed aluminium chain on a rubber string, as in the previous case. Now move two of the vertices of the triangle towards the third one be 0.00000001 units each. Again consider the chain on a level surface and deform it so that its links move towards the interior of the triangle. Geometrically each side of a triangle is deformed into consecutive equal-length chords of a large circle with centre far away from the original triangle.

For $n \geq 5$ this gives an example of $n-3$ interior angles of more than $180^{\circ}$.

To see that $n-2$ or more such angles is not possible, note than the sum of interior angles of any $n$-gon (that does not cut itself) is equal to $(n-2) 180^{\circ}$, but already the sum of the 'large' angles would be greater than that if there were at least $n-2$ of 'large' angles of size greater than $180^{\circ}$.