21st AIME1 2003

1. Find positive integers k, n such that k·n! = (((3!)!)!/3! and n is as large as possible.

2. Concentric circles radii 1, 2, 3, ... , 100 are drawn. The interior of the smallest circle is colored red and the annular regions are colored alternately green and red, so that no two adjacent regions are the same color. Find the total area of the green regions divided by the area of the largest circle.

3. S = {1, 2, 3, 5, 8, 13, 21, 34}. Find ∑ max(A) where the sum is taken over all 28 two-element subsets A of S.

4. Find n such that log₁₀sin x + log₁₀cos x = -1, log₁₀(sin x + cos x) = (log₁₀n - 1)/2.

5. Find the volume of the set of points that are inside or within one unit of a rectangular 3 x 4 x 5 box.

6. Let S be the set of vertices of a unit cube. Find the sum of the areas of all triangles whose vertices are in S.

7. The points A, B, C lie on a line in that order with AB = 9, BC = 21. Let D be a point not on AC such that AD = CD and the distances AD and BD are integral. Find the sum of all possible n, where n is the perimeter of triangle ACD.

8. 0 < a < b < c < d are integers such that a, b, c is an arithmetic progression, b, c, d is a geometric progression, and d - a = 30. Find a + b + c + d.

9. How many four-digit integers have the sum of their two leftmost digits equals the sum of their two rightmost digits?

10. Triangle ABC has AC = BC and ∠ACB = 106^o. M is a point inside the triangle such that ∠MAC = 7^o and ∠MCA = 23^o. Find ∠CMB.

11. The angle x is chosen at random from the interval 0^o < x < 90^o. Find the probability that there is no triangle with side lengths sin²x, cos²x and sin x cos x.

12. ABCD is a convex quadrilateral with AB = CD = 180, perimeter 640, AD ≠ BC, and ∠A = ∠C. Find cos A.

13. Find the number of 1, 2, ... , 2003 which have more 1s than 0s when written in base 2.

14. When written as a decimal, the fraction m/n (with m < n) contains the consecutive digits 2, 5, 1 (in that order). Find the smallest possible n.

15. AB = 360, BC = 507, CA = 780. M is the midpoint of AC, D is the point on AC such that BD bisects ∠ABC. F is the point on BC such that BD and DF are perpendicular. The lines FD and BM meet at E. Find DE/EF.

Answers

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