# Difference between revisions of "2000 AMC 12 Problems"

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== Problem 21 == | == Problem 21 == | ||

+ | Through a point on the [[hypotenuse]] of a [[right triangle]], lines are drawn [[parallel]] to the legs of the triangle so that the triangle is divided into a [[square]] and two smaller right triangles. The area of one of the two small right triangles is <math>m</math> times the area of the square. The [[ratio]] of the area of the other small right triangle to the area of the square is | ||

+ | |||

+ | <math>\text {(A)}\ \frac{1}{2m+1} \qquad \text {(B)}\ m \qquad \text {(C)}\ 1-m \qquad \text {(D)}\ \frac{1}{4m} \qquad \text {(E)}\ \frac{1}{8m^2}</math> | ||

[[2000 AMC 12 Problems/Problem 21|Solution]] | [[2000 AMC 12 Problems/Problem 21|Solution]] |

## Revision as of 02:42, 6 September 2008

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also

## Problem 1

In the year , the United States will host the International Mathematical Olympiad. Let and be distinct positive integers such that the product . What is the largest possible value of the sum ?

## Problem 2

## Problem 3

Each day, Jenny ate of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, remained. How many jellybeans were in the jar originally?

## Problem 4

The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

## Problem 5

If where then

## Problem 6

Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

## Problem 7

How many positive integers have the property that is a positive integer?

## Problem 8

Figures , , , and consist of , , , and non-overlapping squares. If the pattern continued, how many non-overlapping squares would there be in figure ?

## Problem 9

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71,76,80,82, and 91. What was the last score Mrs. Walters entered?

## Problem 10

The point is reflected in the -plane, then its image is rotated by about the -axis to produce , and finally, is translated by 5 units in the positive- direction to produce . What are the coordinates of ?

## Problem 11

Two non-zero real numbers, and satisfy . Which of the following is a possible value of ?

## Problem 12

Let A, M, and C be nonnegative integers such that . What is the maximum value of +++?

## Problem 13

One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

## Problem 14

When the mean, median, and mode of the list

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?

## Problem 15

Let be a function for which . Find the sum of all values of for which .

## Problem 16

## Problem 17

## Problem 18

## Problem 19

## Problem 20

## Problem 21

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square is

## Problem 22

## Problem 23

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from through , inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?