MathDB

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

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 [url=https://artofproblemsolving.com/community/c4h2690808p23355881]Juniors P3, P5.

Problem Statement