19th AIME2 2001

1. Find the largest positive integer such that each pair of consecutive digits forms a perfect square (eg 364).

2. A school has 2001 students. Between 80% and 85% study Spanish, between 30% and 40% study French, and no one studies neither. Find m be the smallest number who could study both, and M the largest number.

3. The sequence a₁, a₂, a₃, ... is defined by a₁ = 211, a₂ = 375, a₃ = 420, a₄ = 523, a_n = a_n-1 - a_n-2 + a_n-3 - a_n-4. Find a₅₃₁ + a₇₅₃ + a₉₇₅.

4. P lies on 8y = 15x, Q lies on 10y = 3x and the midpoint of PQ is (8,6). Find the distance PQ.

5. A set of positive numbers has the triangle property if it has three elements which are the side lengths of a non-degenerate triangle. Find the largest n such that every 10-element subset of {4, 5, 6, ... , n} has the triangle property.

6. Find the area of the large square divided by the area of the small square.

7. The triangle is right-angled with sides 90, 120, 150. The common tangents inside the triangle are parallel to the two sides Find the length of the dashed line joining the centers of the two small circles.

8. The function f(x) satisfies f(3x) = 3f(x) for all real x, and f(x) = 1 - |x-2| for 1 ≤ x ≤ 3. Find the smallest positive x for which f(x) = f(2001).

9. Each square of a 3 x 3 board is colored either red or blue at random (each with probability ½). Find the probability that there is no 2 x 2 red square.

10. How many integers 10ⁱ - 10^j where 0 ≤ j < i ≤ 99 are multiples of 1001?

11. In a tournament club X plays each of the 6 other sides once. For each match the probabilities of a win, draw and loss are equal. Find the probability that X finishes with more wins than losses.

12. The midpoint triangle of a triangle is that obtained by joining the midpoints of its sides. A regular tetrahedron has volume 1. On the outside of each face a small regular tetrahedron is placed with the midpoint triangle as its base, thus forming a new polyhedron. This process is carried out twice more (three times in all). Find the volume of the resulting polyhedron.

13. ABCD is a quadrilateral with AB = 8, BC = 6, BD = 10, ∠A = ∠D and ∠ABD = ∠C. Find CD.

14. Find all the values 0 ≤ θ < 360^o for which the complex number z = cos θ + i sin θ satisfies z²⁸ - z⁸ - 1 = 0.

15. A cube has side 8. A hole with triangular cross-section is bored along a long diagonal. At one vertex it removes the last 2 units of each of the three edges at that vertex. The three sides of the hole are parallel to the long diagonal. Find the surface area of the part of the cube that is left (including the area of the inside of the hole).

Answers

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.