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AMC 12 2007 B

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 3

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

Solution

  
  2020-07-09 06:38:42

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 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 ?

Solution

  
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Question 7

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

Solution

  
  2020-07-09 06:38:42

Question 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  ?

Solution

  
  2020-07-09 06:38:42

Question 9

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

Solution

  
  2020-07-09 06:38:42

Question 10

Some boys and girls are having a car wash to raise money for a class trip to China. Initially % of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then % of the group are girls. How many girls were initially in the group?

Solution

  
  2020-07-09 06:38:42

Question 11

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

Solution

  
  2020-07-09 06:38:42

Question 12

A teacher gave a test to a class in which of the students are juniors and 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?

Solution

  
  2020-07-09 06:38:42

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 14

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

Solution

  
  2020-07-09 06:38:42

Question 15

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

Solution

  
  2020-07-09 06:38:42

Question 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?

Solution

  
  2020-07-09 06:38:42

Question 17

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

Solution

  
  2020-07-09 06:38:42

Question 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 ?

Solution

  
  2020-07-09 06:38:42

Question 19

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

Solution

  
  2020-07-09 06:38:43

Question 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 ?

Solution

  
  2020-07-09 06:38:43

Question 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.)

Solution

  
  2020-07-09 06:38:43

Question 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 ?

Solution

  
  2020-07-09 06:38:43

Question 23

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

Solution

  
  2020-07-09 06:38:43

Question 24

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

Solution

  
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Question 25

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

Solution

  
  2020-07-09 06:38:43

Answer Keys


Question 1: E
Question 2: B
Question 3: D
Question 4: B
Question 5: D
Question 6: D
Question 7: E
Question 8: D
Question 9: A
Question 10: C
Question 11: D
Question 12: C
Question 13: D
Question 14: D
Question 15: E
Question 16: A
Question 17: D
Question 18: C
Question 19: A
Question 20: D
Question 21: A
Question 22: A
Question 23: A
Question 24: A
Question 25: C