Difference between revisions of "2011 AMC 12B Problems"
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+ | {{AMC12 Problems|year=2011|ab=B}} | ||
+ | |||
==Problem 1== | ==Problem 1== | ||
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center> | What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center> | ||
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A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square? | A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square? | ||
− | + | <asy> | |
+ | unitsize(10mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||
+ | dotfactor=4; | ||
+ | pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); | ||
+ | draw(A--B--C--D--E--F--G--H--cycle); | ||
+ | draw(A--D); | ||
+ | draw(B--G); | ||
+ | draw(C--F); | ||
+ | draw(E--H);</asy> | ||
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math> | <math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math> | ||
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==Problem 15== | ==Problem 15== | ||
− | How many positive two- | + | How many positive two-digit integers are factors of <math>2^{24}-1</math>? |
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math> | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math> | ||
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==Problem 16== | ==Problem 16== | ||
+ | |||
+ | Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
+ | |||
+ | Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
+ | |||
+ | A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? | ||
+ | |||
+ | <math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
+ | |||
+ | A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
+ | |||
+ | Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
+ | |||
+ | The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==Problem 22== | ||
+ | |||
+ | Let <math>T_1</math> be a triangle with side lengths <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
+ | |||
+ | A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> and <math>B = (3, -2)</math>. Consider all possible paths of the bug from <math>A</math> to <math>B</math> of length at most <math>20</math>. How many points with integer coordinates lie on at least one of these paths? | ||
+ | |||
+ | <math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
+ | |||
+ | Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter among all the <math>8</math>-sided polygons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3} + 6</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
+ | |||
+ | For every <math>m</math> and <math>k</math> integers with <math>k</math> odd, denote by <math>\left[\frac{m}{k}\right]</math> the integer closest to <math>\frac{m}{k}</math>. For every odd integer <math>k</math>, let <math>P(k)</math> be the probability that | ||
+ | |||
+ | <cmath> \left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right] </cmath> | ||
+ | |||
+ | for an integer <math>n</math> randomly chosen from the interval <math>1 \leq n \leq 99!</math>. What is the minimum possible value of <math>P(k)</math> over the odd integers <math>k</math> in the interval <math>1 \leq k \leq 99</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{44}{87} \qquad \textbf{(D)}\ \frac{34}{67} \qquad \textbf{(E)}\ \frac{7}{13}</math> | ||
+ | |||
+ | [[2011 AMC 12B Problems/Problem 25|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | {{AMC12 box|year=2011|ab=B|before=[[2011 AMC 12A Problems]]|after=[[2012 AMC 12A Problems]]}}{{MAA Notice}} |
Latest revision as of 16:28, 12 July 2020
2011 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
What is
![$\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}?$](http://latex.artofproblemsolving.com/6/1/e/61ee8366705dde051791b9e949cc4eaf30da1215.png)
Problem 2
Josanna's test scores to date are ,
,
,
, and
. Her goal is to raise her test average at least
points with her next test. What is the minimum test score she would need to accomplish this goal?
Problem 3
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid dollars and Bernardo had paid
dollars, where
. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
Problem 4
In multiplying two positive integers and
, Ron reversed the digits of the two-digit number
. His erroneous product was 161. What is the correct value of the product of
and
?
Problem 5
Let be the second smallest positive integer that is divisible by every positive integer less than
. What is the sum of the digits of
?
Problem 6
Two tangents to a circle are drawn from a point . The points of contact
and
divide the circle into arcs with lengths in the ratio
. What is the degree measure of
?
Problem 7
Let and
be two-digit positive integers with mean
. What is the maximum value of the ratio
?
Problem 8
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width meters, and it takes her
seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
Problem 9
Two real numbers are selected independently and at random from the interval . What is the probability that the product of those numbers is greater than zero?
Problem 10
Rectangle has
and
. Point
is chosen on side
so that
. What is the degree measure of
?
Problem 11
A frog located at , with both
and
integers, makes successive jumps of length
and always lands on points with integer coordinates. Suppose that the frog starts at
and ends at
. What is the smallest possible number of jumps the frog makes?
Problem 12
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
Problem 13
Brian writes down four integers whose sum is
. The pairwise positive differences of these numbers are
and
. What is the sum of the possible values of
?
Problem 14
A segment through the focus of a parabola with vertex
is perpendicular to
and intersects the parabola in points
and
. What is
?
Problem 15
How many positive two-digit integers are factors of ?
Problem 16
Rhombus has side length
and
. Region
consists of all points inside of the rhombus that are closer to vertex
than any of the other three vertices. What is the area of
?
Problem 17
Let , and
for integers
. What is the sum of the digits of
?
Problem 18
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Problem 19
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
Problem 20
Triangle has
, and
. The points
, and
are the midpoints of
, and
respectively. Let
be the intersection of the circumcircles of
and
. What is
?
Problem 21
The arithmetic mean of two distinct positive integers and
is a two-digit integer. The geometric mean of
and
is obtained by reversing the digits of the arithmetic mean. What is
?
Problem 22
Let be a triangle with side lengths
, and
. For
, if
and
, and
are the points of tangency of the incircle of
to the sides
, and
, respectively, then
is a triangle with side lengths
, and
, if it exists. What is the perimeter of the last triangle in the sequence
?
Problem 23
A bug travels in the coordinate plane, moving only along the lines that are parallel to the -axis or
-axis. Let
and
. Consider all possible paths of the bug from
to
of length at most
. How many points with integer coordinates lie on at least one of these paths?
Problem 24
Let . What is the minimum perimeter among all the
-sided polygons in the complex plane whose vertices are precisely the zeros of
?
Problem 25
For every and
integers with
odd, denote by
the integer closest to
. For every odd integer
, let
be the probability that
for an integer randomly chosen from the interval
. What is the minimum possible value of
over the odd integers
in the interval
?
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2011 AMC 12A Problems |
Followed by 2012 AMC 12A Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.