MathDB

p1. Determine what is the smallest positive integer by which the number

2010

must be multiplied so that it is a perfect square.
p2. Consider a chessboard of

8\times 8

. An Alba trajectory is defined as any movement that contains

8

white squares, one by one, that touch in a vertex. For example, the white square diagonal is an Alba trajectory. Determine all possible Alba trajectories that can be built on the board.
p3. Find all prime numbers

p

such that

p + 2

and

2p + 5

are also prime.
p4. Find all natural numbers

n

such that

1 / n

has a decimal representation finite.
p5. Prove that all numbers of the form

5^n

, with a positive integer

n

, can be written as the sum of two squares of integer numbers.
p6. Let

ABCD

be a square with side

a

N

be the midpoint of side

BC

and

M

be a point on

CD

such that

MC = 2 MD

. Let

P

be the intersection point of lines

AM

and

DB

Q

be the intersection point of lines

AN

and

BD

. Calculate the area of the pentagon

MPQNC

in terms of

a

.
PS. Juniors P2 was also proposed as [url=https://artofproblemsolving.com/community/c4h2692797p23378693]Seniors P2.

Problem Statement