IMO 1999

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

  1. Problem 1

    Determine all finite sets \(S\) of at least three points in the plane which satisfy the following condition:

    for any two distinct points \(A\) and \(B\) in \(S\), the perpendicular bisector of the line segment \(AB\) is an axis of symmetry for \(S\).

    Solution →

  2. Problem 2

    Let \(n\) be a fixed integer, with \(n \geq 2\).

    (a) Determine the least constant \(C\) such that the inequality \[ \sum_{1\leq i<j\leq n} x_i x_j(x_i^2+x_j^2) \leq C\left(\sum_{1\leq i\leq n}x_i\right)^4 \] holds for all real numbers \(x_1,\cdots,x_n\geq 0\).

    (b) For this constant \(C\), determine when equality holds.

    Solution →

  3. Problem 3

    Consider an \(n \times n\) square board, where \(n\) is a fixed even positive integer. The board is divided into \(n^2\) unit squares. We say that two different squares on the board are adjacent if they have a common side.

    \(N\) unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.

    Determine the smallest possible value of \(N\).

    Solution →

  4. Problem 4

    Determine all pairs \((n,p)\) of positive integers such that

    \(p\) is a prime,
    \(n\) not exceeded \(2p\), and
    \((p-1)^n+1\) is divisible by \(n^{p-1}\).

    Solution →

  5. Problem 5

    Two circles \(G_1\) and \(G_2\) are contained inside the circle \(G\), and are tangent to \(G\) at the distinct points \(M\) and \(N\), respectively. \(G_1\) passes through the center of \(G_2\). The line passing through the two points of intersection of \(G_1\) and \(G_2\) meets \(G\) at \(A\) and \(B\). The lines \(MA\) and \(MB\) meet \(G_1\) at \(C\) and \(D\), respectively.

    Prove that \(CD\) is tangent to \(G_2\).

    Solution →

  6. Problem 6

    Determine all functions \(f:\mathbb{R}\longrightarrow\mathbb{R}\) such that \[ f(x-f(y))=f(f(y))+xf(y)+f(x)-1 \] for all real numbers \(x,y\).

    Solution →

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