SMO 2006 Junior Round 2

Find all integers $x,y$ that satisfy the equation

$$x+y=x^2-xy+y^2.$$

The fraction $\frac 23$ can be expressed as a sum of two distinct unit fractions: $\frac 12+\frac 16$. Show that the fraction $\frac{p-1}{p}$, where $p\ge 5$ is a prime, cannot be expressed as a sum of two distinct unit fractions.

Suppose that each of $n$ people knows exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything he knows, while $B$ tells $A$ nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything?

In $\triangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD=OE$. If $AD\ne AE$, prove that $\angle A=60^\circ$.

You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to make a convex equiangular hexagon (i.e., one whose interior angles are $120^\circ$). Obviously, $n$ cannot be any positive integer. The first three feasible $n$ are $6$, $10$ and $13$. Show that $12$ is not feasible but $14$ is.