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