14th Swedish Mathematical Society Problems 1974

1.  Let an = 2n-1 for n > 0. Let bn = ∑r+s≤n aras. Find bn - bn-1, bn - 2bn-1 and bn.2.  Show that 1 - 1/k ≤ n(k1/n - 1) ≤ k - 1 for all positive integers n and positive reals k. 3.  Let a1 = 1, a2 = 2a1, a3 = 3a2, a4 = 4a3, ... , a9 = 9a8. Find the last two digits of a9. 4.  Find all polynomials p(x) such that p(x2) = p(x)