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Competition Problems

IMO 2001 Problems

Washington, DC, USA
July 1-14, 2001

Download the problems and solutions as: Mathematica NotebookMathematica Notebook     PDFPDF

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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 BCA >= ABC + 30°.

Prove that CAB + COP < 90.


Problem 2

Prove that

for all positive real numbers a, b and c.


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.


Problem 4

Let n be an odd integer greater than 1, and let k1, k2, ..., kn be given integers. For each of the n! permutations a = a1, a2, ..., an of 1, 2, ..., n, let

Prove that there are two permutations b and c, b c, such that n! is a divisor of S(b) - S(c).


Problem 5

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

It is known that BAC = 60° and that AB + BP = AQ + QB.

What are the possible angles of triangle ABC?


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.


View the IMO 2000 problems.

Note: If you do not have a copy of Mathematica, the Mathematica notebook can be read using MathReader, the downloadable notebook reader.