MathDB

p1. The next game is played between two players with a pile of peanuts. The game starts with the man pile being divided into two piles (not necessarily the same size). A move consists of eating all the mana in one pile and dividing the other into two non-empty piles, not necessarily the same size. Players take turns making their moves and the last one wins that I can make a move Show that if the starting number of mana is in at least one of the stacks is even, so the first player can win no matter how his opponent plays (that is, he has a perfect strategy to win).
[url=https://artofproblemsolving.com/community/c4h1847975p12452018]p2. If in the

\vartriangle ABC

, two sides are not greater than their corresponding altitudes, how much do the angles of the triangle measure?
p3. Eight players participated in a free-for-all chess tournament. All players obtained different final scores, the one who finished second having obtained as many points as a total were obtained by the four worst-performing players. What was the result between the players who got fourth and fifth place?
p4. Let

N

be a number of

2002

digits, divisible by

9

. Let

x

be the sum of the digits of

N

. Let

y

the sum of the digits of

x

. Let

z

be the sum of the digits of

y

. Determine the value of

z

.
[url=https://artofproblemsolving.com/community/c4h2916289p26046171]p5. Given a right triangle angle

T

, where the coordinates of its vertices are integer numbers, let

E

be the number of integer coordinate points that belong to the edge of the triangle

T

I

the number of integer coordinate points belonging to the interior of the triangle

T

. Prove that the area

A (T)

of the triangle

T

is given by:

A (T) =\frac{E}{2}+ I

.
p6. Given a

4\times 4

board, determine the value of natural

k

with the following properties:

\bullet

If there are less than

k

tiles on the board, then two rows and two columns can be found such that if they are erased, there will be no pieces left on the board.

\bullet

On the board,

k

tiles can be placed in such a way that when erasing any two rows and two any columns, there will always be at least one tile on the board
[url=https://artofproblemsolving.com/community/c4h1847977p12452026]p7. Given the segment

AB

, let

M

be one point lying on it. Towards the same side of the plane and with base

AM

and

MB

, the squares

AMCD

and

MBEF

are constructed. Let

P

and

Q

be the respective centers of these squares. Determine how the midpoint of the segment

PQ

moves as the point

M

moves along the segment.

Problem Statement