Discrete Mathematics

1.Prove or disprove , that there exist two powers of five whose difference is divisible by 2013


2.Twelve integers are randomly chosen from 1 to 200 inclusively .Prove ,or disprove ,that we can find two disjoint non-empty subsets of the chosen integers such that the two subets give the same sum of elements .

3a.Write a recursive definition for a set which contains all positive even numbers .
3b.a palindrome is a string which reads the same in both directions.Examples of palindrome string are 'dad' ,'civic','level' and 'madam' .
Write a recursive definition for palindrome strings containing Ps and /or Qs .