How to solve problems? - Extreme case

Here are some of my tips, and some problems from MathClub:
Sometimes in a solution of a problem it is crucial to consider extreme objects: The largest number, the closest point or vertex, a degenerate circle, any other limit case.
The minimal counterexample method is a combination of extreme case and a proof by contradiction: Assuming that some statement A is not true, there exists minimal (in some sense) counterexample.If we can somehow "decrease" it, then we get a contradiction.
Problems:
1.Prove that any polyhedron has at least 2 faces with the same number of sides.
2.A traveller went from his city A to the city B of his country, furthest away from A.Then from B he went to the city C, which is furthest away from B, and so on.Prove that if C and A are different cities, then the traveller will never return to A (all distances between cities of this country are different).
3.Numbers are placed on a chessboard.It is given that any number is equal to an arithmetical mean of its neighbours by a side.Prove that all numbers are equal.
4.Prove that for any integer n>1, 1 + 1/2 + 1/3 +...+1/n is not an integer.
5.Let us consider a point P inside a convex polygon and drop the perpendiculars from P to each side or its extension.Prove that at least one of the perpendiculars meets a corresponding side.
6. A six digit number is called lucky if 7 divides the sum of its digits.Are there any 2 consecutive lucky numbers?