SE-SKOLORNAS 1990

Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$ . Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$ .

The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$ . $P$ is chosen to minimise $\sum PA_i$ . Find the minimum.

Find all $a, b$ such that $\sin x + \sin a\ge b \cos x$ for all $x$ .

$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$ . The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$ . Show that $LM = AL + BM$ .

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$ .

Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$ .