Problem 1 Prove that among any 39 consecutive natural numbers it is always possible to find one whose sum of digits is divisible by 11. Problem 2 Sets of 4 positive numbers are made out of each other according to the following rule: (a, b, c, d) (ab, bc, cd, da). Prove that in this (infinite) sequence (a, b, c, d) will never appear again, except when a = b = c = d = 1. Problem 3 Take a series of the numbers 1 and (-1) with a length of 2k (k is natural). The next set is made by multiplying each number by the next one; the last is multiplied by the first. Prove that eventually the set will contain only ones. Problem 4 What is the largest x for which 427 + 41000 + 4x equals the square of a whole number? Problem 5 Prove that for any prime number p > 2 the numerator m of the fraction http://www.geocities.com/CapeCanaveral/Lab/4661/number5.gif is divisible by p.