AustriaMO2011

We have $$\begin{array}{rl}x+y+xy& =3\\ \iff xy+x+y+1& =4\\ \iff (x+1)(y+1)& =4.\end{array}$$ Therefore we can rewrite the inequality in the following way: $$\begin{array}{rl}& x+y\ge 2\\ \iff & x+1+y+1\ge 4\\ \iff & x+1+y+1\ge 2\sqrt{(x+1)(y+1)}.\end{array}$$ The last line is an immediate consequence of the AM-GM inequality. In the last step, equality holds exactly for $x+1=y+1$, i.e. $x=y$. Taking into account the equality $x+y=2$ we obtain $x=y=1$ which is indeed an admissible case of equality and thus the only one.