MathDB

Consider

n

points

A_1,A_2,A_3,\ldots, A_n

(

n\geq 4

) in the plane, such that any three are not collinear. Some pairs of distinct points among

A_1,A_2,A_3,\ldots, A_n

are connected by segments, such that every point is connected with at least three different points. Prove that there exists

k>1

and the distinct points

X_1,X_2,\ldots, X_{2k}

in the set

\{A_1,A_2,A_3,\ldots, A_n\}

, such that for every

i\in \overline{1,2k-1}

the point

X_i

is connected with

X_{i+1}

, and

X_{2k}

is connected with

X_1

Problem Statement