MathDB

p1. Calculate all the solutions

(m, n)

of integers that satisfy the equation

mn^2 = 2009 (n + 1)

.
p2. A cube is cut from side to by a plane that passes through the midpoints of three edges that are on different faces. Determine the plane figure that appears in the cut (the section) and calculate its area.
p3. On a table there are

2009

tokens. Player A removes any number of tokens between

1

and

1004

tokens. Then player

B

removes a non-zero number of tokens no greater than half the number of tokens left by

A

. They continue that way in turns, removing any non-zero numbers of tokens that do not exceed half of the tokens remaining at that time. Loses the player who stays with the last token and cannot, therefore, comply with the rule. Determine which of the two players has a winning strategy.
p4. Consider a parallelogram

W

of vertices

A

B

C

D

and let

P

Q

R

S

the midpoints of the sides

AB

BC

CD

and

DA

respectively. Connecting the points

A

with

R

P

with

C

B

with

S

and

Q

with

D

forms a parallelogram

K

. Calculate the ratio

\frac{area\,\,W}{area\,\,K}

.
p5. Find all the solutions in the real numbers of the system:

x^2 + y^2 + z^2 + w^2 = 4t-4

t^2 + x^2 + y^2 + z^2 = 4w-4

w^2 + t^2 + x^2 + y^2 = 4z-4

z^2 + w^2 + t^2 + x^2 = 4y-4

y^2 + z^2 + w^2 + t^2 = 4x-4

p6. A UN meeting will be attended by

10

representatives from America,

10

from Africa and

10

from Asia. The participants should sit around a round table and, at the beginning of the discussion, ask those whose neighbors come from the same continent to stand up. For example, if an African is sitting between two Asians, then he should get up. The participants have decided to sit in a coordinated manner so that, when asked to stand foot, do it as many as possible. What is said number? Justify your answer.
PS. Juniors P1, P2 were also proposed as [url=https://artofproblemsolving.com/community/c4h2692788p23378606]Seniors P1, P2.

Problem Statement