2000 IMO Problems
Day 1
Problem 1
Two circles and
intersect at two points
and
. Let
be the line tangent to these circles at
and
, respectively, so that
lies closer to
than
. Let
be the line parallel to
and passing through the point
, with
on
and
on
. Lines
and
meet at
; lines
and
meet at
; lines
and
meet at
. Show that
.
Problem 2
Let be positive real numbers with
. Show that
Problem 3
Let be a positive integer and
a positive real number. Initially there are
fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points
and
to the left of
, and letting the flea from
jump over the flea from
to the point
so that
.
Determine all values of such that, for any point
on the line and for any initial position of the
fleas, there exists a sequence of moves that will take them all to the position right of
.
Day 2
Problem 4
A magician has one hundred cards numbered to
. 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 chosen 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 put into a different box.)
Problem 5
Does there exist a positive integer such that
has exactly 2000 prime divisors and
divides
?
Solution
Problem 6
Let ,
, and
be the altitudes of an acute triangle
. The incircle
of triangle
touches the sides
,
, and
at
,
, and
, respectively. Consider the reflections of the lines
,
, and
with respect to the lines
,
, and
. Prove that these images form a triangle whose vertices line on
.