SE-SKOLORNAS 1998

Find all positive integers $a, b, c$ , such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$ .

$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$

A cube side $5$ is made up of unit cubes. Two small cubes are adjacent if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).

$ABCD$ is a quadrilateral with $\angle A = 90o$ , $AD = a$ , $BC = b$ , $AB = h$ , and area $\frac{(a+b)h}{2}$ . What can we say about $\angle B$ ?

Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$ . Show that in any such equation there must be two of the $n$ numbers with a common divisor ( $> 1$ ).

Show that for some $c > 0$ , we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$ .