2000 IMO Problems
Contents
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
.