Difference between revisions of "2000 IMO Problems"
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Revision as of 11:42, 19 April 2024
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
![\[\left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1\]](http://latex.artofproblemsolving.com/e/9/6/e96e8028ff8719b9a661de10c0cb70fc7b43ca67.png)
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 .
