OPM 1995

Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$ . Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?

The diameter $[AC]$ of a circle is divided into four equal segments by points $P, M$ and $Q$ . Consider a segment $[BD]$ that passes through $P$ and cuts the circle at $B$ and $D$ , such that $PD =\frac{3}{2} AP$ . Knowing that the area of the triangle $[ABP]$ has measure $1$ cm $^2$ , calculate the area of $[ABCD]$ ?

Rosa dos Ventos, Aurora Boreal and Manuela do Norte organized a competition between them last weekend, consisting of several athletics events. The winner in each test obtained $x$ points, the second placed $y$ points and the third placed $z$ points ( $x,y,z \in N$ and $x >y>z$ ). The final result of the competition, obtained by adding up the scores in each event, was Rosa had $22$ points, Manuela had $9$ points, Aurora had $9$ points. In how many tests did they compete and who came second in the high jump knowing that the Manuela won the $100$ meters and no one gave up in any race?