SE-SKOLORNAS 1969

Find all integers m, n such that $m^3 = n^3 + n$ .

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$ .

$a_1 \le a_2 \le ... \le a_n$ is a sequence of reals $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le ...\le B_n$ . Show that $\sum a_ib_i \le \sum a_i B_i$ .

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

Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$ , then $N = 0$ mod $3$ .

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?