**AMC 12 Problem Series online course**.

# 2005 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

A scout troop buys candy bars at a price of five for dollars. They sell all the candy bars at the price of two for dollar. What was their profit, in dollars?

## Problem 2

A positive number has the property that of is . What is ?

## Problem 3

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one ﬁfth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

## Problem 4

At the beginning of the school year, Lisa's goal was to earn an A on at least of her quizzes for the year. She earned an A on of the first quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

## Problem 5

An -foot by -foot floor is tiles with square tiles of size foot by foot. Each tile has a pattern consisting of four white quarter circles of radius foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

## Problem 6

In , we have and . Suppose that is a point on line such that lies between and and . What is ?

## Problem 7

What is the area enclosed by the graph of ?

## Problem 8

For how many values of is it true that the line passes through the vertex of the parabola ?

## Problem 9

On a certain math exam, % of the students got points, % got points, % got points, % got points, and the rest got points. What is the difference between the mean and the median score on this exam?

## Problem 10

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?

## Problem 11

An envelope contains eight bills: ones, fives, tens, and twenties. Two bills are drawn at random without replacement. What is the probability that their sum is or more?

## Problem 12

The quadratic equation has roots twice those of , and none of and is zero. What is the value of ?

## Problem 13

Suppose that , , , ... , . What is ?

## Problem 14

A circle having center , with ,is tangent to the lines , and . What is the radius of this circle?

## Problem 15

The sum of four two-digit numbers is . Non of the eight digits is and no two of them are the same. Which of the following is **not** included among the eight digits?

## Problem 16

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

## Problem 17

How many distinct four-tuples of rational numbers are there with

?

## Problem 18

Let and be points in the plane. Define as the region in the first quadrant consisting of those points such that is an acute triangle. What is the closest integer to the area of the region ?

## Problem 19

Let and be two-digit integers such that is obtained by reversing the digits of . The integers and satisfy for some positive integer . What is ?

## Problem 20

Let and be distinct elements in the set

What is the minimum possible value of

## Problem 21

A positive integer has divisors and has divisors. What is the greatest integer such that divides ?

## Problem 22

A sequence of complex numbers is defined by the rule

where is the complex conjugate of and . Suppose that and . How many possible values are there for ?

## Problem 23

Let be the set of ordered triples of real numbers for which

There are real numbers and such that for all ordered triples in we have What is the value of

## Problem 24

All three vertices of an equilateral triangle are on the parabola , and one of its sides has a slope of . The -coordinates of the three vertices have a sum of , where and are relatively prime positive integers. What is the value of ?

## Problem 25

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?