SE-SKOLORNAS 1985

If $a > b > 0$ , prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$

Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2$ times as large as the original number.

Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$ , and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$ . Show that $DE = r$ .

Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge 0$ for all $x$ . Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$ .

In a rectangular coordinate system, $O$ is the origin and $A(a,0)$ , $B(0,b)$ and $C(c,d)$ the vertices of a triangle. Prove that $AB+BC+CA \ge 2CO$ .

X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?