Here are the 6 problems of IMO 2004 (july 12-13 in Athen) Problem 1 ABC is acute angle triangle with AB<>AC. The circle with diameter BC intersects the lines AB and AC respectively at M and N. O is the midpoint of BC. The bisectors of <BAC and <MON intersect at R. Prove that the circumcircles of thev triangles BMR and CNR have a common point lying on the line BC. Problem 2 Find all polynomials f with real coefficients such that, for all reals a,b,c such that ab+bc+ca = 0, we have the relation f(a-b) + f(b-c) + f(c-a) = 2 f(a+b+c) Problem 3 Define a "hook" to be a figure made up of six unit squares as shown in the figure below, or any of the figures obtained by rotations and reflections to this figure. [ ][ ][ ] [ ] [ ] [ ] Determine all mxn rectangles that can be covered without gaps and without overlaps with hooks such that no point of a hook covers area outside the rectangle Problem 4 Let n>=3 be an integer. Let t[1],...,t[n] be positive real numbers such that n^2+1>(t[1]+...+t[n])(1/t[1]+...+1/t[n]) Show that, for all distinct i,j,k, t[i],t[j],t[k] are the side lengths of a triangle Problem 5 In a convex quadrilateral ABCD, the diagonal BD bisects neither <ABC nor <CDA. A point P lies inside ABCD and satisfies <PBC = <DBA and <PDC = <BDA. Prove that ABCD are concyclic if and only if AP = CP. Problem 6 A positive integer is 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. --

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