MathDB

1

Iberoamerican Olympiad 2013 - Problem 5

A

and

B

be two sets such that

A \cup B

is the set of the positive integers, and

A \cap B

is the empty set. It is known that if two positive integers have a prime larger than

2013

as their difference, then one of them is in

A

and the other is in

B

. Find all the possibilities for the sets

A

and

B

.

4

1

Iberoamerican Olympiad 2013 - Problem 4

Let

\Gamma

be a circunference and

O

its center.

AE

is a diameter of

\Gamma

and

B

the midpoint of one of the arcs

AE

of

\Gamma

. The point

D \ne E

in on the segment

OE

. The point

C

is such that the quadrilateral

ABCD

is a parallelogram, with

AB

parallel to

CD

and

BC

parallel to

AD

. The lines

EB

and

CD

meets at point

F

. The line

OF

cuts the minor arc

EB

of

\Gamma

at

I

.
Prove that the line

EI

is the angle bissector of

\angle BEC

.

2

1

Iberoamerican Olympiad 2013 - Problem 2

Let

X

and

Y

be the diameter's extremes of a circunference

\Gamma

and

N

be the midpoint of one of the arcs

XY

of

\Gamma

. Let

A

and

B

be two points on the segment

XY

. The lines

NA

and

NB

cuts

\Gamma

again in

C

and

D

, respectively. The tangents to

\Gamma

at

C

and at

D

meets in

P

. Let

M

the the intersection point between

XY

and

NP

. Prove that

M

is the midpoint of the segment

AB

.

1

Iberoamerican Olympiad 2013 - Problem 1

A set

S

of positive integers is said to be channeler if for any three distinct numbers

a,b,c \in S

, we have

a\mid bc

,

b\mid ca

,

c\mid ab

.
a) Prove that for any finite set of positive integers

\{ c_1, c_2, \ldots, c_n \}

there exist infinitely many positive integers

k

, such that the set

\{ kc_1, kc_2, \ldots, kc_n \}

is a channeler set.
b) Prove that for any integer

n \ge 3

there is a channeler set who has exactly

n

elements, and such that no integer greater than

1

divides all of its elements.

6

1

Beautiful configurations

A beautiful configuration of points is a set of

n

colored points, such that if a triangle with vertices in the set has an angle of at least

120

degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of

n

.

3

1

Product equals sum of squares

Let

A = \{1,...,n\}

with n \textgreater 5. Prove that one can find

B

a finite set of positive integers such that

A

is a subset of

B

and

\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x

2013 IberoAmerican

Subcontests

Iberoamerican Olympiad 2013 - Problem 5

Iberoamerican Olympiad 2013 - Problem 4

Iberoamerican Olympiad 2013 - Problem 2

Iberoamerican Olympiad 2013 - Problem 1

Beautiful configurations

Product equals sum of squares