IR-MO 2008

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab.

A graph is called a self-intersectinggraph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings. a) Find all $n$ 's for which the cycle of length $n$ is self-intersecting. b) Prove that in a self-intersecting graph $|E(G)|\leq|V(G)|$ . c) Find all self-intersecting graphs.

In an old script found in ruins of Perspolis is written:

This script has been finished in a year whose 13th power is
258145266804692077858261512663
You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.

Find the year the script is finished. Give a reason for your answer.