MathDB

A finite set

S{}

n{}

points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed

n{}

-link polylines with vertices at these points will be denoted by

f(S).

Prove that
[*]

f(S)>0

for all sets

S{};

[*]

f(S)=1

if and only if all the points of

S{}

lie on the convex hull of

S{};

[*]if

f(S)>1

then

f(S)\geqslant n-1

, with equality if and only if one point of

S

lies inside the convex hull; [*]if exactly two points of

S{}

lie inside the convex hull, then

f(S)\geqslant\frac{(n-2)(n-3)}{2}.

Let

n\geqslant 3.

Denote by

F(n)

the largest possible value of the function

f(S)

over all admissible sets

S{}

n{}

points. Prove that

F(n)\geqslant3\cdot 2^{(n-8)/3}.

Proposed by E. Bakaev and D. Magzhanov

Problem Statement