From: José H. Nieto <jhnieto@my-deja.com>
Newsgroups: sci.math
Subject: Re: 41st IMO???
Date: Thu, 20 Jul 2000 15:26:39 GMT

In article <8l49lb$hkl$1@salmon.maths.tcd.ie>,
  mdukes@maths.tcd.ie (Mark Dukes) wrote:
> Hello, does anyone have this years IMO problems
> yet or know if they are up on the web already?
> thanx____________Mark
>

1) Let two circles c(1) and c(2) intersect at M and N.
  AB is a direct common tangent (closer to M; A lying on c(1)).
  Thru M draw a line parallel to AB meeting the c(1) at C and c(2)
  at D. NA & NB meet CD at P & Q. E is the intersection of CA&BD.
  Prove that EP=EQ.

2) Let abc=1 (a,b,c are positive reals). Prove that
   (a-1+1/b)(b-1+1/c)(c-1+1/a)<=1.

3) Let t>0 be a real. There are n fleas (n>=2) on a straight line,
  not all at a point. A move means: "a flea A on the left jumps over
  a flea B on its right to go to C s.t. BC/AB =t . Find all t s.t.
  ultimately all fleas can move to the right of M, however far M is.

4) There are 100 cards numbered 1 to 100, and three boxes colored red
  blue and white. Find the number of ways of putting the cards into
  the boxes such that there does not exist a,b belonging to differnt
  boxes and c,d belonging to different boxes (the former pair of boxes
  being not the same as the latter pair of boxes) such that a+b=c+d.

5) Does there exist a positive integer n having exactly 2000 different
  prime divisors s.t. n divides 2^n +1 ?

6) The incircle of ABC touches AB, BC, CA at D, E, F. The foot of the
 altitudes are P, Q, R. Let l(1) be the reflection of QR in EF. Define
 l(2), l(3) similarly. Prove that l(1), l(2), l(3) form a triangle
 whose vertices lie on the incircle.


==============================================================================
problems of recent international mathematics olympiad july 2000 is now
available on the web page

http://animath.free.fr/

( it is written in french)