SE-SKOLORNAS 1994

$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$ . The decimal expansion of $x_1$ has the digit $6$ in place $1994$ . What digit does $x_2$ have in place $1994$ ?

In the triangle $ABC$ , the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$ .

The vertex $B$ of the triangle $ABC$ lies in the plane $P$ . The plane of the triangle meets the plane in a line $L$ . The angle between $L$ and $AB$ is a, and the angle between $L$ and $BC$ is $b$ . The angle between the two planes is $c$ . Angle $ABC$ is $90^o$ . Show that $\sin^2c = \sin^2a + \sin^2b$ .

Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$ .

The polynomial $x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k$ has $k$ distinct real roots. Show that $a_1^2 > \frac{2ka_2}{k-1}$ .

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$ . Also $f(10) = 1$ . How many three digit numbers $n$ satisfy $f(n) = f(N)$ , where $N$ is the "tower" $2, 3, 4, 5$ , in other words, it is $2^a$ , where $a = 3^b$ , where $b = 4^5$ ?