8th Swedish 1968


1. Find the maximum and minimum values of x² + 2y² + 3z² for real x, y, z satisfying x² + y² + z² = 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(m-n), where m, n belong to M cannot exceed n-1.
5. Let a, b be non-zero 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.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.