Difference between revisions of "2007 AMC 12B Problems"
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==Problem 12== | ==Problem 12== | ||
− | A teacher gave a test to a class in which <math>10%</math> of the students are juniors and <math>90%</math> are seniors. The average score on the test was <math>84</math>. The juniors all received the same score, and the average score of the seniors was <math>83</math>. What score did each of the juniors receive on the test? | + | A teacher gave a test to a class in which <math>10\%</math> of the students are juniors and <math>90\%</math> are seniors. The average score on the test was <math>84</math>. The juniors all received the same score, and the average score of the seniors was <math>83</math>. What score did each of the juniors receive on the test? |
<math>\mathrm {(A)} 85\qquad \mathrm {(B)} 88\qquad \mathrm {(C)} 93\qquad \mathrm {(D)} 94\qquad \mathrm {(E)} 98</math> | <math>\mathrm {(A)} 85\qquad \mathrm {(B)} 88\qquad \mathrm {(C)} 93\qquad \mathrm {(D)} 94\qquad \mathrm {(E)} 98</math> |
Revision as of 01:50, 20 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
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
?
![2007 12B AMC-3.png](https://wiki-images.artofproblemsolving.com//8/8a/2007_12B_AMC-3.png)
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 % 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?
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 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?
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
?