2003 CentroAmerican

Part of CentroAmerican

Subcontests

(6)

1

Splitting 8x8 chessboard into groups of 4 squares

A square board with

8\text{cm}

sides is divided into

64

squares square with each side

1\text{cm}

. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side

2\text{cm}

formed by four squares with a common vertex contains two white and two black squares.

1

Triangles have the same perimeter from the two S circles

S_1

S_2

are two circles that intersect at two different points

P

Q

\ell_1

\ell_2

be two parallel lines such that

\ell_1

passes through the point

P

S_1,S_2

A_1,A_2

respectively (both distinct from

P

\ell_2

passes through the point

Q

S_1,S_2

B_1,B_2

respectively (both distinct from

Q

). Show that the triangles

A_1QA_2

B_1PB_2

have the same perimeter.

1

Exponential inequality for integers a,b

a

b

be positive integers with

a>1

b>2

a^b+1\ge b(a+1)

and determine when there is inequality.

1

AH=AG

S

is a circle with

AB

t

is the tangent line to

S

B

. Consider the two points

C

D

t

B

C

D

E

F

are the intersections of

S

AC

AD

G

H

are the intersections of

S

CF

DE

AH=AG

1

A and B take stones from a pile

A

B

take turns playing the following game: There is a pile of

2003

stones. In his first turn,

A

selects a divisor of

2003

and removes this number of stones from the pile.

B

then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

1

About tico numbers

Say a number is tico if the sum of it's digits is a multiple of

2003

.

\text{(i)}

Show that there exists a positive integer

N

such that the first

2003

N,2N,3N,\ldots 2003N

\text{(ii)}

Does there exist a positive integer

N

such that all it's multiples are tico?