OPM 2010

On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$ . Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$ , $E$ , $F$ and $B$ are collinear on this order). Knowing $AE=1$ , find the length of $BF$ .

On each day, more than half of the inhabitants of Évora eats sericaia as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate sericaia as dessert.

Show that any triangle has two sides whose lengths $a$ and $b$ satisfy $\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}$ .