8th AIME 1990

1. The sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers that are not a square or a cube. Find the 500th term.

2. Find (52 + 6√43)^3/2 - (52 - 6√43)^3/2.

3. Each angle of a regular r-gon is 59/58 times larger than each angle of a regular s-gon. What is the largest possible value of s?

4. Find the positive solution to 1/(x²- 10x- 29) + 1/(x²- 10x- 45) = 2/(x²- 10x- 69).

5. n is the smallest positive integer which is a multiple of 75 and has exactly 75 positive divisors. Find n/75.

6. A biologist catches a random sample of 60 fish from a lake, tags them and releases them. Six months later she catches a random sample of 70 fish and finds 3 are tagged. She assumes 25% of the fish in the lake on the earlier date have died or moved away and that 40% of the fish on the later date have arrived (or been born) since. What does she estimate as the number of fish in the lake on the earlier date?

7. The angle bisector of angle A in the triangle A (-8, 5), B (-15, -19), C (1, -7) is ax + 2y + c = 0. Find a and c.

8. 8 clay targets are arranged as shown. In how many ways can they be shot (one at a time) if no target can be shot until the target(s) below it have been shot.

9. A fair coin is tossed 10 times. What is the chance that no two consecutive tosses are both heads.

10. Given the two sets of complex numbers, A = {z : z¹⁸ = 1}, and B = {z : z⁴⁸ = 1}, how many distinct elements are there in {zw : z∈A, w∈B}?

11. Note that 6! = 8·9·10. What is the largest n such that n! is a product of n-3 consecutive positive integers.

12. A regular 12-gon has circumradius 12. Find the sum of the lengths of all its sides and diagonals.

13. How many powers 9ⁿ with 0 ≤ n ≤ 4000 have leftmost digit 9, given that 9⁴⁰⁰⁰ has 3817 digits and that its leftmost digit is 9.

14. ABCD is a rectangle with AB = 13√3, AD = 12√3. The figure is folded along OA and OD to form a tetrahedron. Find its volume.

15. The real numbers a, b, x, y satisfy ax + by = 3, ax² + by² = 7, ax³ + by³ = 16, ax⁴ + by⁴ = 42. Find ax⁵ + by⁵.

Answers

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