2008 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

(8)

1

The old script found in Perspolis

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.

1

Self-intersecting graph

A graph is called a self-intersecting graph 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. http://i35.tinypic.com/x43s5u.png

1

Dividing Mars

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. http://i37.tinypic.com/f2iq8g.png

5

5

6

6

6