# Difference between revisions of "2004 AMC 12A Problems"

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

− | If <math>\sum_{n = 0}^{\infty} | + | If <math>\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5</math>, what is the value of <math>\cos{2\theta}</math>? |

<math>\text {(A)} \frac15 \qquad \text {(B)} \frac25 \qquad \text {(C)} \frac {\sqrt5}{5}\qquad \text {(D)} \frac35 \qquad \text {(E)}\frac45</math> | <math>\text {(A)} \frac15 \qquad \text {(B)} \frac25 \qquad \text {(C)} \frac {\sqrt5}{5}\qquad \text {(D)} \frac35 \qquad \text {(E)}\frac45</math> |

## Revision as of 23:06, 19 August 2012

## 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

Alicia earns dollars per hour, of which is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?

## Problem 2

On the AMC 12, each correct answer is worth points, each incorrect answer is worth points, and each problem left unanswered is worth points. If Charlyn leaves of the problems unanswered, how many of the remaining problems must she answer correctly in order to score at least ?

## Problem 3

For how many ordered pairs of positive integers is ?

## Problem 4

Bertha has daughters and no sons. Some of her daughters have daughters, and the rest have none. Bertha has a total of daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children?

## Problem 5

The graph of the line is shown. Which of the following is true?

## Problem 6

Let , , , , and . Which of the following is the largest?

## Problem 7

A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players , and start with , and tokens, respectively. How many rounds will there be in the game?

## Problem 8

In the overlapping triangles and sharing common side , and are right angles, , , , and and intersect at . What is the difference between the areas of and ?

## Problem 9

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by without altering the volume, by what percent must the height be decreased?

## Problem 10

The sum of consecutive integers is . What is their median?

## Problem 11

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is cents. If she had one more quarter, the average value would be cents. How many dimes does she have in her purse?

## Problem 12

Let and . Points and are on the line , and and intersect at . What is the length of ?

## Problem 13

Let be the set of points in the coordinate plane, where each of and may be , , or . How many distinct lines pass through at least two members of ?

## Problem 14

A sequence of three real numbers forms an arithmetic progression with a first term of . If is added to the second term and is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

## Problem 15

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run meters. They next meet after Sally has run meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?

## Problem 16

The set of all real numbers for which

is defined is . What is the value of ?

## Problem 17

Let be a function with the following properties:

, and

, for any positive integer .

What is the value of ?

## Problem 18

Square has side length . A semicircle with diameter is constructed inside the square, and the tangent to the semicircle from intersects side at . What is the length of ?

## Problem 19

Circles and are externally tangent to each other, and internally tangent to circle . Circles and are congruent. Circle has radius and passes through the center of . What is the radius of circle ?

## Problem 20

Select numbers and between and independently and at random, and let be their sum. Let and be the results when and , respectively, are rounded to the nearest integer. What is the probability that ?

## Problem 21

If , what is the value of ?

## Problem 22

Three mutually tangent spheres of radius rest on a horizontal plane. A sphere of radius rests on them. What is the distance from the plane to the top of the larger sphere?

## Problem 23

A polynomial

has real coefficients with and distinct complex zeroes , with and real, , and

Which of the following quantities can be a nonzero number?

## Problem 24

A plane contains points and with . Let be the union of all disks of radius in the plane that cover . What is the area of ?

## Problem 25

For each integer , let denote the base- number . The product can be expressed as , where and are positive integers and is as small as possible. What is the value of ?