2008 CentroAmerican

Part of CentroAmerican

Subcontests

(6)

1

Independent intersections

ABC

be an acute triangle. Take points

P

Q

AB

AC

, respectively, such that

BPQC

is cyclic. The circumcircle of

ABQ

BC

S

and the circumcircle of

APC

BC

R

PR

QS

intersect again in

L

. Prove that the intersection of

AL

BC

does not depend on the selection of

P

Q

1

p(x) with p(1)=210

Find a polynomial

p\left(x\right)

with real coefficients such that \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right) for all real

x

and p\left(1\right)\equal{}210.

1

Work plan for a week

Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions: a) Each girl will work exactly two days a week b) The 5 assigned couples for the week must be different In how many ways can the girls do the work plan?

1

Frogs game

There are 2008 bags numbered from 1 to 2008, with 2008 frogs in each one of them. Two people play in turns. A play consists in selecting a bag and taking out of it any number of frongs (at least one), leaving

x

x\geq 0

). After each play, from each bag with a number higher than the selected one and having more than

x

frogs, some frogs scape until there are

x

frogs in the bag. The player that takes out the last frog from bag number 1 looses. Find and explain a winning strategy.

1

Parallelograms for a rectangle

ABCD

be a convex cuadrilateral inscribed in a circumference centered at

O

AC

is a diameter. Pararellograms

DAOE

BCOF

are constructed. Show that if

E

F

lie on the circumference then

ABCD

is a rectangle.

1

Sum of the digits of N and 2N

Find the least positive integer

N

such that the sum of its digits is 100 and the sum of the digits of

2N