IMO 2001 Problems

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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°.

Solution

Problem 2

Prove that

for all positive real numbers a, b and c.

Solution

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

Problem 4

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

Solution

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?

Solution

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

View the IMO 2000 problems.

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