p1. Find all prime numbers $p$ such that $2p + p^2$ is a prime number. p2. Given a triangle $ABC$, acute angle at $B$ and $C$, show that there exists a only point $D$ on $BC$ such that segment $EF$ is parallel to side $BC$, where $E$ and $F$ are the intersection points of the perpendiculars from point $D$ to sides $AB$ and $AC$ respectively. p3. Of a total of $49$ small white squares of a board of $7\times 7$ have been painted $29$ black. Show that there always exists at least one square of $2\times 2$ with at least three little black squares. p4. Consider a triangle $\vartriangle ABC$, and a point $P$ inside it. When drawing the lines $AP$, $BP$ and $CP$, the intersection points $D$, $E$ and $F$ are determined on the sides $BC$, $CA$ and $AB$ respectively. The triangle is divided into 6 triangles ($\vartriangle AFP$, $\vartriangle FPB$, $\vartriangle BDP$, $\vartriangle DPC$, $\vartriangle CPE$, $\vartriangle EPA$). Show that if $4$ of these triangles have the same area then points $D$, $E$, $F$ are the midpoints of the respective sides. p5. Prove that the number $(36a + b) (a + 36b)$ is never a power of $2$, for any choice of natural numbers $a$ and $b$. p6. In a group of $2015$ people the following is observed: for each pair of people who know each other, between the two they know everyone, but they do not have acquaintances in common. Prove that it is possible to separate people into two groups, such that in each group no one knows. Clarification: In this problem, if $A$ knows $B$, then we also have that $B$ knows $A$, that is, knowing oneself is a symmetric relationship. PS. Seniors P1, P4 were also proposed as Juniors P3, P5.