SE-SKOLORNAS 1968

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

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

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?

For $n\ne 0$ , 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 $n-1$ .

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 $r$ , $m(a, b) \le r < 0$ for all $a, b$ .