OME 1990

Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to $\sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$

and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}$ .

Every point of the plane is painted with one of three colors. Can we always find two points a distance $1$ cm apart which are of the same color?

Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$ is independent of $a$ for $a \ge - \frac{3}{4}$ and evaluate it.

On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$ , where $0 < p < 1$ is variable. (a) Find the area of triangle $A' B' C'$ in terms of $p$ . (b) Find the value of $p$ which minimizes this area. (c) Find the locus of the intersection point $P$ of the lines through $A'$ and $C'$ parallel to $AB$ and $AC$ respectively.

There are $n$ points in the plane so that no two pairs are equidistant. Each point is connected to the nearest point by a segment. Show that no point is connected to more than five points.