SE-SKOLORNAS 1991

Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$ .

$x, y$ are positive reals such that $x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}$ . Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$ .

The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$ , $x_{k+1} = [(n - \sum_0^k x_i)/2]$ . Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$ .

$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$ . A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$ .

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$ s than $n^2$ .

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.