BALKAN 2002 Problem 1

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 $X_1$ .