IMO 2003

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

  1. Problem 1

    \(S\) is the set \(\{1,2,3,\ldots,1000000\}\). Show that for any subset \(A\) of \(S\) with \(101\) elements we can find \(100\) distinct elements \(x_i\) of \(S\), such that the sets \(\{a+x_i\mid a\in A\}\) are all pairwise disjoint.

    Solution →

  2. Problem 2

    Find all pairs \((m,n)\) of positive integers such that \(\frac{m^2}{2mn^2-n^3+1}\) is a positive integer.

    Solution →

  3. Problem 3

    A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is \(\sqrt{3}/2\) times the sum of their lengths Show that all the hexagon’s angles are equal.

    Solution →

  4. Problem 4

    \(ABCD\) is cyclic. The feet of the perpendicular from \(D\) to the lines \(AB,BC,CA\) are \(P,Q,R\) respectively. Show that the angle bisectors of \(ABC\) and \(CDA\) meet on the line \(AC\) iff \(RP=RQ\).

    Solution →

  5. Problem 5

    Given \(n>2\) and reals \(x_1\le x_2\le\cdots\le x_n\), show that \[ \left(\sum_{i,j}|x_i-x_j|\right)^2 \le \frac{2}{3}(n^2-1)\sum_{i,j}(x_i-x_j)^2. \] Show that we have equality iff the sequence is an arithmetic progression.

    Solution →

  6. Problem 6

    Show that for each prime \(p\), there exists a prime \(q\) such that \(n^p-p\) is not divisible by \(q\) for any positive integer \(n\).

    Solution →

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