IMO 2001 ProblemsWashington, DC, USA
July 1-14, 2001
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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°.
for all positive real numbers a, b and c.
Twenty-one girls and twenty-one 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.
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).
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?
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.