MathDB

2

13rd ibmo - rep. dominicana 1998/q3.

n

with the following property: between any collection of

n

distinct natural numbers in the set

\{1,2, \dots,999\}

it is possible to choose four different

a,\ b,\ c,\ d

such that:

a + 2b + 3c = d

.

13rd ibmo - rep. dominicana 1998/q6.

Let

\lambda

the positive root of the equation

t^2-1998t-1=0

. It is defined the sequence

x_0,x_1,x_2,\ldots,x_n,\ldots

by x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots Find the remainder of the division of

x_{1998}

by

1998

. Note:

\lfloor{x}\rfloor

is the greatest integer less than or equal to

x

.

2

13rd ibmo - rep. dominicana 1998/q2.

The circumference inscribed on the triangle

ABC

is tangent to the sides

BC

,

CA

and

AB

on the points

D

,

E

and

F

, respectively.

AD

intersect the circumference on the point

Q

. Show that the line

EQ

meet the segment

AF

at its midpoint if and only if

AC=BC

.

13rd ibmo - rep. dominicana 1998/q5.

Find the maximal possible value of

n

such that there exist points

P_1,P_2,P_3,\ldots,P_n

in the plane and real numbers

r_1,r_2,\ldots,r_n

such that the distance between any two different points

P_i

and

P_j

is

r_i+r_j

.

1

2

13rd ibmo - rep. dominicana 1998/q4.

There are representants from

n

different countries sit around a circular table (

n\geq2

), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every

n

, the maximal number of people that can be sit around the table.

13rd ibmo - rep. dominicana 1998/q1.

Given 98 points in a circle. Mary and Joseph play alternatively in the next way: - Each one draw a segment joining two points that have not been joined before. The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.

1998 IberoAmerican

Subcontests

13rd ibmo - rep. dominicana 1998/q3.

13rd ibmo - rep. dominicana 1998/q6.

13rd ibmo - rep. dominicana 1998/q2.

13rd ibmo - rep. dominicana 1998/q5.

13rd ibmo - rep. dominicana 1998/q4.

13rd ibmo - rep. dominicana 1998/q1.