IMO 2004

6 problems with complete solutions · Official problem paper (PDF)

  1. Problem 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 \(\angle BAC\) and \(\angle MON\) intersect at \(R\). Prove that the circumcircles of the triangles \(BMR\) and \(CNR\) have a common point lying on the side \(BC\).

    Solution →

  2. 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 following relations \[ f(a-b)+f(b-c)+f(c-a)=2f(a+b+c). \]

    Solution →

  3. Problem 3

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

    ┌───┬───┬───┐
│   │   │   │
├───┼───┼───┤
│   │   │   │
├───┤   └───┘
│   │
└───┘

    Determine all \(m\times n\) rectangles that can be covered without gaps and without overlaps with hooks such that

    • the rectangle is covered without gaps and without overlaps
    • no part of a hook covers area outside the rectagle.

    Solution →

  4. Problem 4

    Let \(n \geq 3\) be an integer. Let \(t_1,t_2,\ldots,t_n\) be positive real numbers such that \[ n^2+1>(t_1+t_2+\ldots+t_n) \left(\frac1{t_1}+\frac1{t_2}+\ldots+\frac1{t_n}\right). \] Show that \(t_i,t_j,t_k\) are side lengths of a triangle for all \(i,j,k\) with \(1\leq i<j<k\leq n\).

    Solution →

  5. Problem 5

    In a convex quadrilateral \(ABCD\) the diagonal \(BD\) does not bisect the angles \(ABC\) and \(CDA\). The point \(P\) lies inside \(ABCD\) and satisfies \[ \angle PBC=\angle DBA \quad\text{and}\quad \angle PDC=\angle BDA. \] Prove that \(ABCD\) is a cyclic quadrilateral if and only if \(AP=CP\).

    Solution →

  6. 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.

    Solution →

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