p1. For how many natural numbers $n$, the numbers $n, 1 + 2n, 1 + 4n$ are all prime? p2. Find all the solutions in the real numbers of the following system: $x + y + z = 1 $ $x^2 + y^2 + z^2 = 1 $ $x^3 + y^3 + z^3 = 1$ p3. Consider a triangle $\vartriangle ABC$, right at $C$ and a point $P$ at a distance $4$ from vertex $A$, $7$ from vertex $B$, and $1$ from vertex $C$. What are the smallest and largest lengths that side $AC$ can have? p4. Let $ j, k$ be two positive integers. A pile of pencils, all of different colors, is distributed first in $j$ drawers and then the same pile in $k$ drawers. Determine the minimum number of pencils needed in the stack to guarantee that no matter how they are distributed, there will be two colors together in the same drawer both times. PS. Problem 1 was also used as Juniors p1.