IMO 2001 ProblemsWashington, DC, USAJuly 114, 2001 Download the problems and solutions as: Mathematica Notebook PDF
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Problem 1Let ABC be an acuteangled 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 2Prove that
for all positive real numbers a, b and c. Problem 3Twentyone girls and twentyone boys took part in a mathematical contest.
Prove that there was a problem that was solved by at least three girls and at least three boys. Problem 4Let n be an odd integer greater than 1, and let k_{1}, k_{2}, ..., k_{n} be given integers. For each of the n! permutations a = a_{1}, a_{2}, ..., a_{n} 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 5In 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 6Let 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.
