IMO 2002

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

  1. Problem 1

    \(S\) is the set of all \((h,k)\) with \(h,k\) non-negative integers such that \(h+k<n\). Each element of \(S\) is colored red or blue, so that if \((h,k)\) is red and \(h'\le h,k'\le k\), then \((h',k')\) is also red. A type 1 subset of \(S\) has \(n\) blue elements with different first member and a type 2 subset of \(S\) has \(n\) blue elements with different second member. Show that there are the same number of type 1 and type 2 subsets.

    Solution →

  2. Problem 2

    \(BC\) is a diameter of a circle center \(O\). \(A\) is any point on the circle with \(\angle AOC > 60^\circ\). \(EF\) is the chord which is the perpendicular bisector of \(AO\). \(D\) is the midpoint of the minor arc \(AB\). The line through \(O\) parallel to \(AD\) meets \(AC\) at \(J\). Show that \(J\) is the incenter of triangle \(CEF\).

    Solution →

  3. Problem 3

    Find all pairs of integers \(m>2,n>2\) such that there are infinitely many positive integers \(k\) for which \(k^n+k^2-1\) divides \(k^m+k-1\).

    Solution →

  4. Problem 4

    The positive divisors of the integer \(n > 1\) are \(d_1 < d_2 < \ldots < d_k\), so that \(d_1 = 1, d_k = n\). Let \(d = d_1d_2 + d_2d_3 + \cdots + d_{k-1}d_k\). Show that \(d < n^2\) and find all \(n\) for which \(d\) divides \(n^2\).

    Solution →

  5. Problem 5

    Find all real-valued functions on the reals such that \((f(x)+f(y))((f(u)+f(v))=f(xu-yv)+f(xv+yu)\) for all \(x,y,u,v\).

    Solution →

  6. Problem 6

    \(n>2\) circles of radius \(1\) are drawn in the plane so that no line meets more than two of the circles. Their centers are \(O_1,O_2,\cdots,O_n\). Show that \[ \sum_{i<j}1/O_iO_j\le (n-1)\pi/4. \]

    Solution →

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