MathDB

p1. From a box full of coins, n coins are extracted with which to they form

n

towers of one coin each. The only move allowed in the game is stack two different towers into one and then add an extra coin from the box to tower. Is there some

n

such that you can finish the game with a single tower of

2014

coins?
[url=https://artofproblemsolving.com/community/c4h1845739p12426604]p2. The points

P,Q,R

are the midpoints of the sides

BC,CD

and

DA

of a rectangle

ABCD

respectively and

M

is the midpoint of the segment

QR

. The area of the rectangle is

320

. Calculate the area of the triangle

APM

.
p3. Consider a chessboard of

n \times n

squares, with at least one corner black. Determine all values of

n

for for which a row starting at a white square can go through all the white squares passing exactly one time for each of them.
p4. Determine the units digit of the following number:

(1 + 1^2) + (2 + 2^2) + (3 + 3^2) +...+ (2014 + 2014^2)

p5. In a group of people, each one votes for someone else or abstains. If

A

votes for

B

B

votes for

C

C

votes for

D

and

D

votes for A we say that there was a collusion of four people. Similarly, a collusion of

n

people is defined,

n\ge 2

. In this vote there were no collusions and there was at least one vote. Show that there is at least one person who voted and who did not receive a vote and who exists at least a person who abstained and did receive votes.
[url=https://artofproblemsolving.com/community/c4h2917778p26063766]p6. Prove that if a quadrilateral

ABCD

can be cut into a finite number of parallelograms, then

ABCD

is a parallelogram.
PS. Problem 6 was also proposed as Seniors P6.

Problem Statement