

1. Let a_{n} = 2^{n1} for n > 0. Let b_{n} = ∑_{r+s≤n} a_{r}a_{s}. Find b_{n}  b_{n1}, b_{n}  2b_{n1} and b_{n}.


2. Show that 1  1/k ≤ n(k^{1/n}  1) ≤ k  1 for all positive integers n and positive reals k.


3. Let a_{1} = 1, a_{2} = 2^{a1}, a_{3} = 3^{a2}, a_{4} = 4^{a3}, ... , a_{9} = 9^{a8}. Find the last two digits of a_{9}.


4. Find all polynomials p(x) such that p(x^{2}) = p(x)^{2} for all x. Hence find all polynomials q(x) such that q(x^{2}  2x) = q(x2)^{2}.


5. Find the smallest positive real t such that x_{1} + x_{3} = 2t x_{2}, x_{2} + x_{4} = 2t x_{3}, x_{3} + x_{5} = 2t x_{4} has a solution x_{1}, x_{2}, x_{3}, x_{4}, x_{5} in nonnegative reals, not all zero.


6. For which n can we find positive integers a_{1}, a_{2}, ... , a_{n} such that a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2} is a square?

