

1. Find all positive integers a, b, c, such that (8a5b)^{2} + (3b2c)^{2} + (3c7a)^{2} = 2.


2. ABC is a triangle. Show that c ≥ (a+b) sin(C/2).


3. A cube side 5 is made up of unit cubes. Two small cubes are adjacent if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).


4. ABCD is a quadrilateral with ∠A = 90^{o}, AD = a, BC = b, AB = h, and area (a+b)h/2. What can we say about ∠B?


5. Show that for any n > 5 we can find positive integers x_{1}, x_{2}, ... , x_{n} such that 1/x_{1} + 1/x_{2} + ... + 1/x_{n} = 1997/1998. Show that in any such equation there must be two of the n numbers with a common divisor (> 1).


6. Show that for some c > 0, we have 2^{1/3}  m/n > c/n^{3} for all integers m, n with n ≥ 1.

