

1. Find the maximum and minimum values of x^{2} + 2y^{2} + 3z^{2} for real x, y, z satisfying x^{2} + y^{2} + z^{2} = 1.


2. How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ... , 6?


3. Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?


4. For n ≠ 0, let f(n) be the largest k such that 3^{k} divides n. If M is a set of n > 1 integers, show that the number of possible values for f(mn), where m, n belong to M cannot exceed n1.


5. Let a, b be nonzero integers. Let m(a, b) be the smallest value of cos ax + cos bx (for real x). Show that for some r, m(a, b) ≤ r < 0 for all a, b.

