MathDB

p1. Let

a

be a positive integer. Prove that the equation

x^2-y^2 = a^3

always has solutions

x, y

in the integers.
p2. A reporter arrived or after a

100

-meter dash ended. To write his article, got the following information from five people who saw the arrival of the race:

\bullet

Alejandro: Juan came second and Oscar came third.

\bullet

Boris: Pedro came third and Tom came fifth.

\bullet

Cristian: Tom thus came first and Pedro came second.

\bullet

Jorge: Juan came second and Raul came fourth.

\bullet

Diego: Oscar came first and Raul came fourth. It was clear to the reporter that he did not have the correct information, so he consulted with a colleague, but he only mentioned that in each answer there was a correct and an incorrect position. Help the reporter to find the correct order of arrival.
p3.

ABCDEFGH

is a cube with side

2

. Let

M

be the midpoint of

BC

and

N

the midpoint of

EF

. Find the area of the quadrilateral

AMHN

.
p4. Given a trapezoid

ABCD

, with

AB \parallel CD

and

AD = DC =\frac{AB}{2}

.Determine

\angle ACB

.
p5. Are there two positive integers

a, b

whose sum is

1995

and whose product is a multiple of nineteen ninety five?
p6. A rectangle

ABCD

is divided into

5

rectangles

R_1, R_2, R_3, R_4

and

R_5

, as indicated in the figure. It is known that

R_5

is a square and that the areas of

R_1, R_2, R_3

and

R_4

are equal to each other. Prove that

ABCD

is a square. ([color=#f00]missing figure)
p7. Suppose that in a room no boy dances with all the girls, but that each girl dances with at least one boy. Prove that there are two boys:

m

and

m'

, and two girls,

n

and

n'

such that

m

dances with

n

, and

m'

dances with

n'

, but

m

does not dance with

n'

, and

m'

does not dance with

n

Problem Statement