SE-SKOLORNAS 1964

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$ .

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$ . The point $P$ is chosen to minimise $\max (PH_i)$ . Find the largest possible value of $\max (PH_i)$ for $n = 3$ . Find the best upper bound you can for $n = 4$ .

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$ . We seek estimates of $a_1$ . If $n = 2$ , find the smallest and largest possible values of $a_1$ . Find corresponding estimates for other values of $n$ .