First Day
Taejon, Korean, July 19, 2000
Duration: 4 hr 30 min
7 points each problem
Problem 1.
Two circles
1 and
2
intersect at M and N .
Let 
be the common tangent to
1 and
2
so that M is closer to
than N
is. Let
touch
1
at A and
2
at B . Let the line through M
parallel to
meet the circle
1
again at C and the circle
2
again at D.
Lines CA and DB meet at E;
lines AN and
CD meet at P; lines BN and
CD meet at Q.
Show that EP = EQ.
Problem 2.
Let a, b, c be positive real
numbers such that abc = 1. Prove that
Problem 3.
Let n
2 be
a positive integer. Initially, there
are n fleas on a horizontal
line, not all at the same point.
For a positive real number, 
define a move as follows;
choose any two fleas, at points A and
B , with A to the left of B;
let the flea at A jump to the point
C on the line to the right of with BC/AB
=
.
Determine all values of
such that, for any
point M on the line
and any initial positions of the n
fleas. There is a finite sequence of moves that will take all
the fleas to positions to the right of M .
Second
Day
Taejon, Korean, July 20, 2000
Duration:
4 hr 30 min
7
points each problem
Problem
4
A
magician has one hundred cards numbered 1 to 100. He
puts them into three boxes, a red one, a white one and a blue
one, so that each box contains at least one card.
A
member of the audience selects two of the three boxes, chooses
one card from each and announces the sum of the numbers on the
shosen cards. Given this sum, the magician identifies the box
from which no card has been chosen.
How many ways are there to put all the
cards into the boxes so that this trick always works? (Two
ways are considered different if at least one card is into a
different box.)
Problem 5.
Determine whether or not there exists a
positive integrer n such that
n is divisible by exactly 2000 different prime
numbers, and 
is divisible by n.
Problem
6.
Let 
be the altitudes of an acute-angled triangle ABC.
The incircle of the triangle ABC
touches the sides BC,
CA, AB at 
respectively. Let the lines 
be the reflections of the lines 
in the lines 
respectively.
Prove that determine a triangle whose vertices lie on the incircle of the
triangle ABC.
