NO-ABEL 2004

(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$ . (b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$ , prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$

In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$ , the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$ . (a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle. (b) Find the area of quadrilateral $ABCD$ .

Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends. (a) Show that the chief's order can be achieved by using $n^2/4$ different types of stones. (b) Prove that this is not necessarily true with less than $n^2/4$ types.