IMO 2004

From: Ignacio Larrosa Caņestro (ilarrosaQUITARMAYUSCULAS_at_mundo-r.com)
Date: 07/14/04 

Date: Wed, 14 Jul 2004 22:13:32 +0200

>From a mail list,

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. 

-- 
Best regards,
Ignacio Larrosa Caņestro
A Coruņa (Espaņa)
ilarrosaQUITARMAYUSCULAS@mundo-r.com

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