2009 CentroAmerican

Part of CentroAmerican

Subcontests

(6)

1

Easy calculation

P

be the product of all non-zero digits of the positive integer

n

. For example, P(4) \equal{} 4, P(50) \equal{} 5, P(123) \equal{} 6, P(2009) \equal{} 18. Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).

1

Numbers in a circle

We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different. a) Is it possible to place the numbers from 1 to 2009 satisfying this property b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property

1

Diophantine Equation

Find all prime numbers

p

q

such that p^3 \minus{} q^5 \equal{} (p \plus{} q)^2.

1

Stones and Boxes Game

There are 2009 boxes numbered from 1 to 2009, some of which contain stones. Two players,

A

B

, play alternately, starting with

A

. A move consists in selecting a non-empty box

i

, taking one or more stones from that box and putting them in box i \plus{} 1. If i \equal{} 2009, the selected stones are eliminated. The player who removes the last stone wins a) If there are 2009 stones in the box 2 and the others are empty, find a winning strategy for either player. b) If there is exactly one stone in each box, find a winning strategy for either player.

1

Parallel Lines

Given an acute and scalene triangle

ABC

H

be its orthocenter,

O

its circumcenter,

E

F

the feet of the altitudes drawn from

B

C

, respectively. Line

AO

intersects the circumcircle of the triangle again at point

G

FE

BC

X

Y

respectively. Let

Z

be the point of intersection of line

AH

and the tangent line to the circumcircle at

G

HX

YZ

1

Cyclic Quadrilateral

\item Two circles

\Gamma_1

\Gamma_2

intersect at points

A

B

. Consider a circle

\Gamma

\Gamma_1

\Gamma_2

, which is tangent to both of them at

D

E

respectively. Let

C

be one of the intersection points of line

AB

\Gamma

F

be the intersection of line

EC

\Gamma_2

G

be the intersection of line

DC

\Gamma_1

H

I

be the intersection points of line

ED

\Gamma_1

\Gamma_2

respectively. Prove that

F

G

H

I

are on the same circle.