18th AIME1 2000

1. Find the smallest positive integer n such that if 10ⁿ = M·N, where M and N are positive integers, then at least one of M and N must contain the digit 0.

2. m, n are integers with 0 < n < m. A is the point (m, n). B is the reflection of A in the line y = x. C is the reflection of B in the y-axis, D is the reflection of D in the x-axis, and E is the reflection of D in the y-axis. The area of the pentagon ABCDE is 451. Find u + v.

3. m, n are relatively prime positive integers. The coefficients of x² and x³ in the expansion of (mx + b)²⁰⁰⁰ are equal. Find m + n.

4. The figure shows a rectangle divided into 9 squares. The squares have integral sides and adjacent sides of the rectangle are coprime. Find the perimeter of the rectangle.

5. Two boxes contain between them 25 marbles. All the marbles are black or white. One marble is taken at random from each box. The probability that both marbles are black is 27/50. If the probability that both marbles are white is m/n, where m and n are relatively prime, find m + n.

6. How many pairs of positive integers m, n have n < m < 1000000 and their arithmetic mean equal to their geometric mean plus 2?

7. x, y, z are positive reals such that xyz = 1, x + 1/z = 5, y + 1/x = 29. Find z + 1/y.

8. A sealed conical vessel is in the shape of a right circular cone with height 12, and base radius 5. The vessel contains some liquid. When it is held point down with the base horizontal the liquid is 9 deep. How deep is it when the container is held point up and base horizontal?

9. Find the real solutions to: log₁₀(2000xy) - log₁₀x log₁₀y = 4, log₁₀(2yz) - log₁₀y log₁₀z = 1, log₁₀zx - log₁₀z log₁₀x = 0.

10. The sequence x₁, x₂, ... , x₁₀₀ has the property that, for each k, x_k is k less than the sum of the other 99 numbers. Find x₅₀.

11. Find [S/10], where S is the sum of all numbers m/n, where m and n are relatively prime positive divisors of 1000.

12. The real-valued function f on the reals satisfies f(x) = f(398 - x) = f(2158 - x) = f(3214 - x). What is the largest number of distinct values that can appear in f(0), f(1), f(2), ... , f(999)?

13. A fire truck is at the intersection of two straight highways in the desert. It can travel at 50mph on the highway and at 14mph over the desert. Find the area it can reach in 6 mins.

14. Triangle ABC has AB = AC. P lies on AC, and Q lies on AB. We have AP = PQ = QB = BC. Find angle ACB/angle APQ.

15. There are cards labeled from 1 to 2000. The cards are shuffled and placed in a pile. The top card is placed on the table, then the next card at the bottom of the pile. Then the next card is placed on the table to the right of the first card, and the next card is placed at the bottom of the pile. This process is continued until all the cards are on the table. The final order (from left to right) is 1, 2, 3, ... , 2000. In the original pile, how many cards were above card 1999?

Answers

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