IMO Problems 2004

IMO Problems 2004

Day 1

IMO 2004 Problem 1

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

IMO 2004 Problem 2

Find all polynomials f with real coefficients such that for all real a, b, and c such that ab + bc + ca = 0, we have the following relation
f(a – b) + f(b – c) + f(c – a) = 2f(a + b + c).

IMO 2004 Problem 3

Define a hook to be a figure made up of 6 unit squares, as shown in the picture or any of the figures obtained by applying rotations and reflections to this figure.

IMO 2004 Problem 3

Determine all m × n rectangles that can be covered without gaps and without overlaps with hook such that
(a) the rectangle is covered without gaps and without overlaps.
(b) no part of a hook covers area outside the rectangle.

Day 2

IMO 2004 Problem 4

Let n ≥ 3, be an integer. Let a₁, a₂, a₃, …,aₙ be positive real numbers such that

n² + 1 > (a₁ + a₂ + a₃ + … + aₙ)(1/a₁ + 1/a₂ + 1/a₃ + … + 1/aₙ)

Show that aᵢ, aⱼ, aₖ are side lengths of a triangle for all i, j, and k with 1 ≤ i < j < k ≤ n.

IMO 2004 Problem 5

In a convex quadrilateral ABCD, the diagonal BD bisects neither the angle(ABC) nor the angle(CDA). The 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.

IMO 2004 Problem 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.