16th AIME 1998

1. For how many k is lcm(6⁶, 8⁸, k) = 12¹²?

2. How many ordered pairs of positive integers m, n satisfy m ≤ 2n ≤ 60, n ≤ 2m ≤ 60?

3. The graph of y² + 2xy + 40|x| = 400 divides the plane into regions. Find the area of the bounded region.

4. Nine tiles labeled 1, 2, 3, ... , 9 are randomly divided between three players, three tiles each. Find the probability that the sum of each player's tiles is odd.

5. Find |A₁₉ + A₂₀ + ... + A₉₈|, where A_n = ½n(n-1) cos(n(n-1)½π).

6. ABCD is a parallelogram. P is a point on the ray DA such that PQ = 735, QR = 112. Find RC.

7. Find the number of ordered 4-tuples (a, b, c, d) of odd positive integers with sum 98.

8. The sequence 1000, n, 1000-n, n-(1000-n), ... terminates with the first negative term (the n+2th term is the nth term minus the n+1th term). What positive integer n maximises the length of the sequence?

9. Two people arrive at a cafe independently at random times between 9am and 10am and each stay for m minutes. What is m if there is a 40% chance that they are in the cafe together at some moment.

10. 8 sphere radius 100 rest on a table with their centers at the vertices of a regular octagon and each sphere touching its two neighbors. A sphere is placed in the center so that it touches the table and each of the 8 spheres. Find its radius.

11. A cube has side 20. Two adjacent sides are UVWX and U'VWX'. A lies on UV a distance 15 from V, and F lies on VW a distance 15 from V. E lies on WX' a distance 10 from W. Find the area of intersection of the cube and the plane through A, F, E.

12. ABC is equilateral, D, E, F are the midpoints of its sides. P, Q, R lie on EF, FD, DE respectively such that A, P, R are collinear, B, Q, P, are collinear, and C, R, Q are collinear. Find area ABC/area PQR.

13. Let A be any set of positive integers, so the elements of A are a₁ < a₂ < ... < a_n. Let f(A) = ∑ a_k i^k. Let S_n = ∑ f(A), where the sum is taken over all non-empty subsets A of {1, 2, ... , n}. Given that S₈ = -176-64i, find S₉.

14. An a x b x c box has half the volume of an (a+2) x (b+2) x (c+2) box, where a ≤ b ≤ c. What is the largest possible c?

15. D is the set of all 780 dominos [m,n] with 1≤m<n≤40 (note that unlike the familiar case we cannot have m = n). Each domino [m,n] may be placed in a line as [m,n] or [n,m]. What is the longest possible line of dominos such that if [a,b][c,d] are adjacent then b = c?

Answers

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