IMO 2005

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

  1. Problem 1

    Six points are chosen on the sides of an equilateral triangle \(ABC\): \(A_1, A_2\) on \(BC\), \(B_1, B_2\) on \(CA\) and \(C_1, C_2\) on \(AB\), such that they are the vertices of a convex hexagon \(A_1A_2B_1B_2C_1C_2\) with equal side lengths. Prove that the lines \(A_1B_2\), \(B_1C_2\) and \(C_1A_2\) are concurrent.

    Solution →

  2. Problem 2

    Let \(a_1,a_2,\ldots\) be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer \(n\) the numbers \(a_1,a_2,\ldots,a_n\) leave \(n\) different remainders upon division by \(n\). Prove that every integer occurs exactly once in the sequence \(a_1,a_2,\ldots\).

    Solution →

  3. Problem 3

    Let \(x,y,z\) be three positive reals such that \(xyz\geq 1\). Prove that \[ \frac{x^5-x^2}{x^5+y^2+z^2} +\frac{y^5-y^2}{x^2+y^5+z^2} +\frac{z^5-z^2}{x^2+y^2+z^5}\geq 0. \]

    Solution →

  4. Problem 4

    Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n-1,\quad n\geq 1. \]

    Solution →

  5. Problem 5

    Let \(ABCD\) be a fixed convex quadrilateral with \(BC = DA\) and \(BC\) not parallel with \(DA\). Let two variable points \(E\) and \(F\) lie of the sides \(BC\) and \(DA\), respectively and satisfy \(BE = DF\). The lines \(AC\) and \(BD\) meet at \(P\), the lines \(BD\) and \(EF\) meet at \(Q\), the lines \(EF\) and \(AC\) meet at \(R\).

    Prove that the circumcircles of the triangles \(PQR\), as \(E\) and \(F\) vary, have a common point other than \(P\).

    Solution →

  6. Problem 6

    In a mathematical competition, in which 6 problems were posed to the participants, every two of these problems were solved by more than \(\frac{2}{5}\) of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.

    Solution →

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