Date: 14 Jul 2004 05:06:46 +0100
From: Adrian Sanders
Subject: Re: IMO

Here are the IMO problems:

Day 1:

1. Let ABC be an acute-angled triangle with AB /neq AC. The circle with 
diameter BC intersects the sides AB and AC at M and N, respectively. Denote 
by O the midpoint of the side BC. The bisectors of the angles BAC and MON 
intersect at R. Prove that the circumcircles of the triangles BMR and CNR 
have a common point lying on the side BC.

2. Find all polynomials P(x) with real coefficients which satisfy the 
equality

P(a-b) + P(b-c) + P(c-a) = P(a+b+c)

for all real numbers a,b,c such that ab + bc + ca = 0.

3. (This question doesn't really lend itself to my writing it in this email 
- I'll have a go...) Define a _hook_ to be a figure made up of six unit 
squares in the plane looking like

{(0,0), (0,1), (0,2), (1,2), (2,2), (2,1)}

(These are the coordinates of the six squares in the hook - on the paper it 
just gives the picture)

or any of the figures obtained by applying rotations and reflections to 
this figure. Determine all m*n rectangles that can be tiled by hooks.

Day 2:

4. Let n \geq 3 be an integer. Let t_1, t_2, ... , t_n be positive real 
numbers such that

n^2 + 1 > (t_1 + t_2 + ... + t_n)(1/t_1 + 1/t_2 + ... + 1/t_n).

Show that t_i, t_j, t_k are side lengths of a triangle for all i,j,k with 1 
\leq i < j < k \leq n.

5. In a convex quadrilateral ABCD the diagonal BD bisects neither the angle 
ABC nor the angle CDA. A point P lies inside ABCD and satisfies

angle PBC = angle DBA  and  angle PDC = angle BDA.

Prove that ABCD is a cyclic quadrilateral if and only if AP = CP.

6. We call a positive integer _alternating_ if every two consecutive digits 
in its decimal representation are of different parity. Find all positive 
integers n such that n has a multiple which is alternating.


Coordination begins this morning, and I should be able to give bulletins 
from now on. I won't say anything yet about how our team and others have 
found the problems, because I imagine many of you will want to have a go at 
them.

We have performed solidly - and in case you are wondering, Paul does not 
have a 42 possibility. Six medals looks realistic for us at this stage.

Best wishes,

Adrian


On Jun 30 2004, Adrian Sanders wrote:

> Dear all,
> 
> As you probably know, the UK team is setting off for the IMO this 
> Saturday. I shall do my best to keep you informed of our performance. The 
> papers are on Monday 12th and Tuesday 13th; so unless we end up in an 
> almighty co-ordination row, the results should be known late on the 
> evening of Thursday 15th.
> 
> Many, many thanks to you all for your assistance in the team's 
> preparations this year - I'm sure our illustrious six will do you proud.
> 
> 
> Best wishes, Adrian