# Difference between revisions of "2007 AMC 12B Problems"

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==Problem 18== | ==Problem 18== | ||

− | Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. | + | Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. The three-digit integer <math>abc</math> lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer <math>acb</math> lies two thirds of the way between the same two squares. What is <math>a+b+c</math>? |

<math>\mathrm {(A)} 10</math> <math>\mathrm {(B)} 13</math> <math>\mathrm {(C)} 16</math> <math>\mathrm {(D)} 18</math> <math>\mathrm {(E)} 21</math> | <math>\mathrm {(A)} 10</math> <math>\mathrm {(B)} 13</math> <math>\mathrm {(C)} 16</math> <math>\mathrm {(D)} 18</math> <math>\mathrm {(E)} 21</math> |

## Revision as of 13:13, 12 November 2007

## 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

## Problem 1

Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?

## Problem 2

A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?

## Problem 3

The point is the center of the circle circumscribed about triangle , with and , as shown. What is the degree measure of ?

## Problem 4

At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?

## Problem 5

The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?

## Problem 6

Triangle has side lengths , , and . Two bugs start simultaneously from and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point . What is ?

## Problem 7

All sides of the convex pentagon are of equal length, and . What is the degree measure of ?

## Problem 8

Tom's age is years, which is also the sum of the ages of his three children. His age years ago was twice the sum of their ages then. What is ?

## Problem 9

A function has the property that for all real numbers . What is ?

## Problem 10

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40%$ (Error compiling LaTeX. ! Missing $ inserted.) of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30%$ (Error compiling LaTeX. ! Missing $ inserted.) of the group are girls. How many girls were initially in the group?

## Problem 11

The angles of quadrilateral satisfy . What is the degree measure of , rounded to the nearest whole number?

## Problem 12

A teacher gave a test to a class in which $10%$ (Error compiling LaTeX. ! Missing $ inserted.) of the students are juniors and $90%$ (Error compiling LaTeX. ! Missing $ inserted.) are seniors. The average score on the test was . The juniors all received the same score, and the average score of the seniors was . What score did each of the juniors receive on the test?

## Problem 13

A traffic light runs repeatedly through the following cycle: green for seconds, then yellow for seconds, and then red for seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?

## Problem 14

Point is inside equilateral . Points , , and are the feet of the perpendiculars from to , , and , respectively. Given that , , and , what is ?

## Problem 15

The geometric series has a sum of , and the terms involving odd powers of have a sum of . What is ?

## Problem 16

Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?

## Problem 17

If is a nonzero integer and is a positive number such that , what is the median of the set ?

## Problem 18

Let , , and be digits with . The three-digit integer lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer lies two thirds of the way between the same two squares. What is ?

## Problem 19

Rhombus , with side length , is rolled to form a cylinder of volume by taping to . What is ?

## Problem 20

The parallelogram bounded by the lines , , , and has area . The parallelogram bounded by the lines , , , and has area . Given that , , , and are positive integers, what is the smallest possible value of ?

## Problem 21

The first positive integers are each written in base . How many of these base- representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

## Problem 22

Two particles move along the edges of equilateral in the direction starting simultaneously and moving at the same speed. One starts at , and the other starts at the midpoint of . The midpoint of the line segment joining the two particles traces out a path that encloses a region . What is the ratio of the area of to the area of ?

## Problem 23

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to times their perimeters?

## Problem 24

How many pairs of positive integers are there such that and is an integer?

## Problem 25

Points and are located in 3-dimensional space with and . The plane of is parallel to . What is the area of ?