p1. The figure shows the triangle $ABC$, right at $C$, its circle circumscribed and semicircles built on the two legs. If $AB = 5$, $AC = 4$, and $BC = 3$, find the sum of the areas of the two shaded regions. p2. Find the greatest power of $3$ that divides $10^{2012} - 1$. p3. In how many different ways can a board of $2 \times 20$ squares be covered by $1\times 1$, with $2 \times 1$ domino pieces, so that the pieces do not overlap or stick out of the board? p4. In a certain game there are several piles of stones that can be modified according to the following rules: a) Two of the piles can be put together into one. b) If a pile has an even number of stones, it can be divided into two piles with the same number of stones each. At the beginning there are three piles, one of them has $5$ stones, another has $49$ and another has $51$. Determine if it is possible to achieve, with successive movements, and following rules a) and b), that at the end there are $105$ piles, each one with a stone. p5. Each vertex of a cube is assigned the value $+1$ or $-1$, and each face the product of the values assigned to its vertices. What values can the sum of the $14$ numbers thus obtained, have? p6. The quadrilateral $ABCD$ in the figure is a trapezoid, and we have $EF\parallel AB\parallel CD$. Furthermore, $EF$ passes through point $G$, where the diagonals $AC$ and $BD$ intersect. The lengths of $AB$ and $CD$ are known to be, respectively, $ 12$ and $4$. Find the length of $EF$. PS. Juniors P1,P3 were also posted as Seniors P3, P4.