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