# 2004 AMC 12B Problems

## 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

At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?

## Problem 2

In the expression , the values of , , , and are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?

## Problem 3

If and are positive integers for which , what is the value of ?

## Problem 4

An integer , with , is to be chosen. If all choices are equally likely, what is the probability that at least one digit of is a 7?

## Problem 5

On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had Canadian dollars left. What is the sum of the digits of ?

## Problem 6

Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?

## Problem 7

A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?

## Problem 8

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?

## Problem 9

The point is rotated clockwise around the origin to point . Point is then reflected over the line to point . What are the coordinates of ?

## Problem 10

An annulus is the region between two concentric circles. The concentric circles in the ﬁgure have radii and , with . Let be a radius of the larger circle, let be tangent to the smaller circle at , and let be the radius of the larger circle that contains . Let , , and . What is the area of the annulus?

## Problem 11

All the students in an algebra class took a -point test. Five students scored , each student scored at least , and the mean score was . What is the smallest possible number of students in the class?

## Problem 12

In the sequence , , , , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is . What is the term in this sequence?

## Problem 13

If and with and real, what is the value of ?

## Problem 14

In , , , and . Points and lie on and , respectively, with . Points and are on so that and are perpendicular to . What is the area of pentagon ?

## Problem 15

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?

## Problem 16

A function is defined by , where and is the complex conjugate of . How many values of satisfy both and ?

## Problem 17

For some real numbers and , the equation has three distinct positive roots. If the sum of the base- logarithms of the roots is , what is the value of ?

## Problem 18

Points and are on the parabola , and the origin is the midpoint of . What is the length of ?

## Problem 19

A truncated cone has horizontal bases with radii and . A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?

## Problem 20

Each face of a cube is painted either red or blue, each with probability . The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

## Problem 21

The graph of is an ellipse in the first quadrant of the -plane. Let and be the maximum and minimum values of over all points on the ellipse. What is the value of ?

## Problem 22

The square

is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of ?

## Problem 23

The polynomial has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of are possible?

## Problem 24

In , , and is an altitude. Point is on the extension of such that . The values of , , and form a geometric progression, and the values of form an arithmetic progression. What is the area of ?

## Problem 25

Given that is a -digit number whose first digit is , how many elements of the set have a first digit of ?