2007 IberoAmerican

Part of IberoAmerican

Subcontests

(6)

1

Family of hexagons

\mathcal{F}

be a family of hexagons

H

satisfying the following properties: i)

H

has parallel opposite sides. ii) Any 3 vertices of

H

can be covered with a strip of width 1. Determine the least

\ell\in\mathbb{R}

such that every hexagon belonging to

\mathcal{F}

can be covered with a strip of width

\ell

. Note: A strip is the area bounded by two parallel lines separated by a distance

\ell

. The lines belong to the strip, too.

1

Dragon moving in a board

19\times 19

board, a piece called dragon moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The draconian distance between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let

C

be a corner square, and

V

the square neighbor of

C

that has only a point in common with

C

. Show that there exists a square

X

of the board, such that the draconian distance between

C

X

is greater than the draconian distance between

C

V

1

Atresvido numbers

Let's say a positive integer

n

is atresvido if the set of its divisors (including 1 and

n

) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

1

Flag strategy at a circle

A

B

, fight for a territory limited by a circumference.

A

n

B

n

n\geq 2

, fixed). They play alternatively and

A

begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved. Once all

2n

flags have been placed, territory is divided between the two teams. A point of the territory belongs to

A

if the closest flag to it is blue, and it belongs to

B

if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from

A

B

). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas. Prove that, for every

n

B

has a winning strategy.

1

Circle centered at I

ABC

be a triangle with incenter

I

\Gamma

be a circle centered at

I

, whose radius is greater than the inradius and does not pass through any vertex. Let

X_{1}

be the intersection point of

\Gamma

AB

B

X_{2}

X_{3}

the points of intersection of

\Gamma

BC

X_{2}

B

X_{4}

be the point of intersection of

\Gamma

CA

C

K

be the intersection point of lines

X_{1}X_{2}

X_{3}X_{4}

AK

bisects segment

X_{2}X_{3}

1

Sequence with ceiling function

Given an integer

m

, define the sequence

\left\{a_{n}\right\}

as follows: a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1 Find all values of

m

a_{2007}

is the first integer appearing in the sequence. Note: For a real number

x

\left\lceil x\right\rceil

is defined as the smallest integer greater or equal to

x

. For example, \left\lceil\pi\right\rceil\equal{}4, \left\lceil 2007\right\rceil\equal{}2007.