Difference between revisions of "2008 AMC 12B Problems"
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([[2008 AMC 12B Problems/Problem 10|Solution]]) | ([[2008 AMC 12B Problems/Problem 10|Solution]]) | ||
==Problem 11== | ==Problem 11== | ||
+ | A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top <math>\frac{1}{8}</math> of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet? | ||
+ | |||
+ | <math>\textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4-\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000</math> | ||
([[2008 AMC 12B Problems/Problem 11|Solution]]) | ([[2008 AMC 12B Problems/Problem 11|Solution]]) | ||
==Problem 12== | ==Problem 12== | ||
+ | For each positive integer <math>n</math>, the mean of the first <math>n</math> terms of a sequence is <math>n</math>. What is the <math>2008</math>th term of the sequence? | ||
+ | |||
+ | <math>\textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4030056 \qquad \textbf{(E)}\ 4032064</math> | ||
([[2008 AMC 12B Problems/Problem 12|Solution]]) | ([[2008 AMC 12B Problems/Problem 12|Solution]]) | ||
==Problem 13== | ==Problem 13== | ||
+ | Vertex <math>E</math> of equilateral triangle <math>\triangle ABE</math> is in the interior of unit square <math>ABCD</math>. Let <math>R</math> be the region consisting of all points inside <math>ABCD</math> and outside <math>\triangle ABE</math> whose distance from <math>\overline{AD}</math> is between <math>\frac{1}{3}</math> and <math>\frac{2}{3}</math>. What is the area of <math>R</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{12-5\sqrt{3}}{72} \qquad \textbf{(B)}\ \frac{12-5\sqrt{3}}{36} \qquad \textbf{(C)}\ \frac{\sqrt{3}}{18} \qquad \textbf{(D)}\ \frac{3-\sqrt{3}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{12}</math> | ||
([[2008 AMC 12B Problems/Problem 13|Solution]]) | ([[2008 AMC 12B Problems/Problem 13|Solution]]) | ||
==Problem 14== | ==Problem 14== | ||
+ | A circle has a radius of <math>\log_{10}{(a^2)}</math> and a circumference of <math>\log_{10}{(b^4)}</math>. What is <math>\log_{a}{b}</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{4\pi} \qquad \textbf{(B)}\ \frac{1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}</math> | ||
([[2008 AMC 12B Problems/Problem 14|Solution]]) | ([[2008 AMC 12B Problems/Problem 14|Solution]]) | ||
Line 116: | Line 128: | ||
([[2008 AMC 12B Problems/Problem 16|Solution]]) | ([[2008 AMC 12B Problems/Problem 16|Solution]]) | ||
==Problem 17== | ==Problem 17== | ||
+ | Let <math>A</math>, <math>B</math> and <math>C</math> be three distinct points on the graph of <math>y=x^2</math> such that line <math>AB</math> is parallel to the <math>x</math>-axis and <math>\triangle ABC</math> is a right triangle with area <math>2008</math>. What is the sum of the digits of the <math>y</math>-coordinate of <math>C</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 20</math> | ||
([[2008 AMC 12B Problems/Problem 17|Solution]]) | ([[2008 AMC 12B Problems/Problem 17|Solution]]) | ||
==Problem 18== | ==Problem 18== | ||
A pyramid has a square base <math>ABCD</math> and vertex <math>E</math>. The area of square <math>ABCD</math> is <math>196</math>, and the areas of <math>\triangle ABE</math> and <math>\triangle CDE</math> are <math>105</math> and <math>91</math>, respectively. What is the volume of the pyramid? | A pyramid has a square base <math>ABCD</math> and vertex <math>E</math>. The area of square <math>ABCD</math> is <math>196</math>, and the areas of <math>\triangle ABE</math> and <math>\triangle CDE</math> are <math>105</math> and <math>91</math>, respectively. What is the volume of the pyramid? | ||
− | |||
<math>\textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt {6} \qquad \textbf{(C)}\ 392\sqrt {2} \qquad \textbf{(D)}\ 392\sqrt {3} \qquad \textbf{(E)}\ 784</math> | <math>\textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt {6} \qquad \textbf{(C)}\ 392\sqrt {2} \qquad \textbf{(D)}\ 392\sqrt {3} \qquad \textbf{(E)}\ 784</math> |
Revision as of 10:53, 29 February 2008
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
A basketball player made baskets during a game. Each basket was worth either
or
points. How many different numbers could represent the total points scored by the player?
(Solution)
Problem 2
A block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
(Solution)
Problem 3
A semipro baseball league has teams with players each. League rules state that a player must be paid at least
dollars, and that the total of all players' salaries for each team cannot exceed
dollars. What is the maximum possiblle salary, in dollars, for a single player?
(Solution)
Problem 4
On circle , points
and
are on the same side of diameter
,
, and
. What is the ratio of the area of the smaller sector
to the area of the circle?
(Solution)
Problem 5
A class collects dollars to buy flowers for a classmate who is in the hospital. Roses cost
dollars each, and carnations cost
dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly
dollars?
(Solution)
Problem 6
Postman Pete has a pedometer to count his steps. The pedometer records up to steps, then flips over to
on the next step. Pete plans to determine his mileage for a year. On January
Pete sets the pedometer to
. During the year, the pedometer flips from
to
forty-four times. On December
the pedometer reads
. Pete takes
steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
(Solution)
Problem 7
For real numbers and
, define
. What is
?
(Solution)
Problem 8
Points and
lie on
. The length of
is
times the length of
, and the length of
is
times the length of
. The length of
is what fraction of the length of
?
(Solution)
Problem 9
Points and
are on a circle of radius
and
. Point
is the midpoint of the minor arc
. What is the length of the line segment
?
(Solution)
Problem 10
Bricklayer Brenda would take hours to build a chimney alone, and bricklayer Brandon would take
hours to build it alone. When they work together they talk a lot, and their combined output is decreased by
bricks per hour. Working together, they build the chimney in
hours. How many bricks are in the chimney?
(Solution)
Problem 11
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
(Solution)
Problem 12
For each positive integer , the mean of the first
terms of a sequence is
. What is the
th term of the sequence?
(Solution)
Problem 13
Vertex of equilateral triangle
is in the interior of unit square
. Let
be the region consisting of all points inside
and outside
whose distance from
is between
and
. What is the area of
?
(Solution)
Problem 14
A circle has a radius of and a circumference of
. What is
?
(Solution)
Problem 15
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let be the region formed by the union of the square and all the triangles, and
be the smallest convex polygon that contains
. What is the area of the region that is inside
but outside
?
(Solution)
Problem 16
A rectangular floor measures by
feet, where
and
are positive integers with
. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width
foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair
?
(Solution)
Problem 17
Let ,
and
be three distinct points on the graph of
such that line
is parallel to the
-axis and
is a right triangle with area
. What is the sum of the digits of the
-coordinate of
?
(Solution)
Problem 18
A pyramid has a square base and vertex
. The area of square
is
, and the areas of
and
are
and
, respectively. What is the volume of the pyramid?
(Solution)
Problem 19
A function is defined by
for all complex numbers
, where
and
are complex numbers and
. Suppose that
and
are both real. What is the smallest possible value of
(Solution)
Problem 20
Michael walks at the rate of feet per second on a long straight path. Trash pails are located every
feet along the path. A garbage truck travels at
feet per second in the same direction as Michael and stops for
seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
(Solution)
Problem 21
Two circles of radius 1 are to be constructed as follows. The center of circle is chosen uniformly and at random from the line segment joining
and
. The center of circle
is chosen uniformly and at random, and independently of the first choice, from the line segment joining
to
. What is the probability that circles
and
intersect?
(Solution)
Problem 22
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
(Solution)
Problem 23
The sum of the base- logarithms of the divisors of
is
. What is
?
(Solution)
Problem 24
Let . Distinct points
lie on the
-axis, and distinct points
lie on the graph of
. For every positive integer
,
is an equilateral triangle. What is the least
for which the length
?
(Solution)
Problem 25
Let be a trapezoid with
,
,
,
, and
. Bisectors of
and
meet at
, and bisectors of
and
meet at
. What is the area of hexagon
?
(Solution)
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