Here are three “little” problems that my friend John (from UK) gave me yesterday.

(1) How many 5-digit numbers can be constructed using only 1, 2 and 3 such that each of those three digits is used at least once?

(2) Find all pairs of positive integers (m,n) such that 1 + 5\cdot 2^m = n^2.

(3) Find all positive integers a, b such that a^b = b^a.