

1. Let {x} denote the fractional part of x = x  [x]. The sequences x_{1}, x_{2}, x_{3}, ... and y_{1}, y_{2}, y_{3}, ... are such that lim {x_{n}} = lim {y_{n}} = 0. Is it true that lim {x_{n} + y_{n}} = 0? lim {x_{n}  y_{n}} = 0?


2. a_{1} + a_{2} + ... + a_{n} = 0, for some k we have a_{j} ≤ 0 for j ≤ k and a_{j} ≥ 0 for j > k. If a_{i} are not all 0, show that a_{1} + 2a_{2} + 3a_{3} + ... + na_{n} > 0.


3. Show that an integer = 7 mod 8 cannot be sum of three squares.


4. Let f(x) = 1 + 2/x. Put f_{1}(x) = f(x), f_{2}(x) = f(f_{1}(x)), f_{3}(x) = f(f_{2}(x)), ... . Find the solutions to x = f_{n}(x) for n > 0.


5. Let f(r) be the number of lattice points inside the circle radius r, center the origin. Show that lim_{r→∞} f(r)/r^{2} exists and find it. If the limit is k, put g(r) = f(r)  kr^{2}. Is it true that lim_{r→∞} g(r)/r^{h} = 0 for any h < 2?

