IMO 2001

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

  1. Problem 1

    Let \(ABC\) be an acute-angled triangle with circumcentre \(O\). Let \(P\) on \(BC\) be the foot of the altitude from \(A\).

    Suppose that \(\angle BCA \ge \angle ABC + 30^\circ\).

    Prove that \(\angle CAB + \angle COP < 90^\circ\).

    Solution →

  2. Problem 2

    Prove that \[ \frac{a}{\sqrt{a^2+8bc}}+ \frac{b}{\sqrt{b^2+8ca}}+ \frac{c}{\sqrt{c^2+8ab}}\geq 1 \] for all positive real numbers \(a\), \(b\) and \(c\).

    Solution →

  3. Problem 3

    Twenty-one girls and twenty-one boys took part in a mathematical contest.

    • Each contestant solved at most six problems.
    • For each girl and each boy, at least one problem was solved by both of them.

    Prove that there was a problem that was solved by at least three girls and at least three boys.

    Solution →

  4. Problem 4

    Let \(n\) be an odd integer greater than 1, and let \(k_1,k_2,\ldots,k_n\) be given integers. For each of the \(n!\) permutations \(a=(a_1,a_2,\ldots,a_n)\) of \(1,2,\ldots,n\), let \[ S(a)=\sum_{i=1}^{n}k_i a_i. \] Prove that there are two permutations \(b\) and \(c\), \(b\ne c\), such that \(n!\) is a divisor of \(S(b)-S(c)\).

    Solution →

  5. Problem 5

    In a triangle \(ABC\), let \(AP\) bisect \(\angle BAC\), with \(P\) on \(BC\), and let \(BQ\) bisect \(\angle ABC\), with \(Q\) on \(CA\).

    It is known that \(\angle BAC=60^\circ\) and that \(AB+BP=AQ+QB\).

    What are the possible angles of triangle \(ABC\)?

    Solution →

  6. Problem 6

    Let \(a,b,c,d\) be integers with \(a>b>c>d>0\). Suppose that \[ ac+bd=(b+d+a-c)(b+d-a+c). \] Prove that \(ab+cd\) is not prime.

    Solution →

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