

1. Find all integers m, n such that m^{3} = n^{3} + n.


2. Show that tan π/3n is irrational for all positive integers n.


3. a_{1} ≥ a_{2} ≥ ... ≥ a_{n} is a sequence of reals. b_{1}, b_{2}, b_{3}, ... b_{n} is any rearrangement of the sequence B_{1} ≥ B_{2} ≥ ... ≥ B_{n}. Show that ∑ a_{i}b_{i} ≤ &sum a_{i}B_{i}.


4. Define g(x) as the largest value of y^{2}  xy for y in [0, 1]. Find the minimum value of g (for real x).


5. Let N = a_{1}a_{2} ... a_{n} in binary. Show that if a_{1}  a_{2} + a_{3}  ... + (1)^{n1}a_{n} = 0 mod 3, then N = 0 mod 3.


6. Given 3n points in the plane, no three collinear, is it always possible to form n triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

