2002 IberoAmerican

Part of IberoAmerican

Subcontests

(3)

2

17th ibmo - el salvador 2002/q3

P

be a point in the interior of the equilateral triangle

\triangle ABC

\sphericalangle{APC}=120^\circ

M

be the intersection of

CP

AB

N

the intersection of

AP

BC

. Find the locus of the circumcentre of the triangle

MBN

P

17th ibmo - el salvador 2002/q6.

A policeman is trying to catch a robber on a board of

2001\times2001

squares. They play alternately, and the player whose trun it is moves to a space in one of the following directions:

\downarrow,\rightarrow,\nwarrow

.
If the policeman is on the square in the bottom-right corner, he can go directly to the square in the upper-left corner (the robber can not do this). Initially the policeman is in the central square, and the robber is in the upper-left adjacent square. Show that:

a)

The robber may move at least

10000

times before the being captured.

b)

The policeman has an strategy such that he will eventually catch the robber.
Note: The policeman can catch the robber if he reaches the square where the robber is, but not if the robber enters the square occupied by the policeman.

2

17th ibmo - el salvador 2002./q2

Given any set of

9

points in the plane such that there is no

3

of them collinear, show that for each point

P

of the set, the number of triangles with its vertices on the other

8

points and that contain

P

on its interior is even.

17th ibmo - el salvador 2002./q5.

The sequence of real numbers

a_1,a_2,\dots

is defined as follows:

a_1=56

a_{n+1}=a_n-\frac{1}{a_n}

n\ge 1

. Show that there is an integer

1\leq{k}\leq2002

a_k<0

2

17th ibmo - el salvador 2002.

The integer numbers from

1

2002

are written in a blackboard in increasing order

1,2,\ldots, 2001,2002

. After that, somebody erases the numbers in the

(3k+1)-th

(1,4,7,\dots)

. After that, the same person erases the numbers in the

(3k+1)-th

positions of the new list (in this case,

2,5,9,\ldots

). This process is repeated until one number remains. What is this number?

17th ibmo - el salvador 2002/q4.

\triangle{ABC}

with all its sides of different length,

D

AC

BD

is the angle bisector of

\sphericalangle{ABC}

E

F

, respectively, be the feet of the perpendicular drawn from

A

C

BD

M

be the point on

BC

DM

is perpendicular to

BC

\sphericalangle{EMD}=\sphericalangle{DMF}