MathDB

p1. Consider a grid board of

16\times 16

cells. What is the maximum number of lockers that can be painted black so that are there no more than

8

black boxes in each row, each column and each diagonal of the board?
p2. On a blackboard,

4

points

A

B

C

and

D

are still drawn so that they form an convex quadrilateral. Find a point

E

inside the quadrilateral of so that the sum of the distances to the

4

vertices is the smallest possible.
p3. Find all the prime numbers

p

such that

2^p + p^2

is a prime number.
p4. Determine if the number

1! + 2! + ... + 2015!

is a perfect square.
p5. 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.
p6. Sebastian and Fernando are preparing to play the following game: at a table there are

2015

tokens, which are red on one side and black on the other. Initially the tokens are randomly flipped and played alternately in turns. On each shift, it is allowed remove any non-zero quantity of tokens of the same color or flip any quantity not null of sheets of the same color. Whoever removes the last token wins. If Sebastian plays first, who has a winning strategy?
PS. Juniors P3, P5 were also proposed as [url=https://artofproblemsolving.com/community/c4h2693883p23392778]Seniors P1, P4.

Problem Statement