1998 Cono Sur Olympiad

Part of Cono Sur Olympiad

Subcontests

(6)

1

the world will end because of goblins in Terra Brasilis (corrected)

In Terra Brasilis there are

n

n

goblins live, each in a house. There are one-way routes such that: - each route joins two houses, - in each house exactly one route begins, - in each house exactly one route ends. If a route goes from house

A

B

, then we will say that house

B

is next to house

A

. This relationship is not symmetric, that is: in this situation, not necessarily house

A

is next to house

B

. Every day, from day

1

, each goblin leaves the house where he is and arrives at the next house. A legend of Terra Brasilis says that when all the goblins return to the original position, the world will end. a) Show that the world will end. b) If

n = 98

, show that it is possible for elves to build and guide the routes so that the world does not end before

300,000

1

no of stops in a bus line

The mayor of a city wishes to establish a transport system with at least one bus line, in which: - each line passes exactly three stops, - every two different lines have exactly one stop in common, - for each two different bus stops there is exactly one line that passes through both. Determine the number of bus stops in the city.

1

How many ways???

98

cards, in each one we will write one of the numbers:

1, 2, 3, 4,...., 97, 98

. We can order the

98

cards, in a sequence such that two consecutive numbers

X

Y

X - Y

is greater than

48

, determine how and how many ways we can make this sequence!!

1

Only prove that XY = BC

H

be the orthocenter of the triangle

ABC

M

is the midpoint of the segment

BC

X

be the point of the intersection of the line

HM

BC

A

) of the circumcircle of

ABC

Y

be the point of intersection of the line

BH

with the circle, show that

XY = BC

1

Find all functions

Find all functions

R-->R

f(x^2) - f(y^2) + 2x + 1 = f(x + y)f(x - y)

1

30% of the natural numbers

Prove that, least

30

% of the natural numbers

n

1

1000000

the first digit of

2^n

1