Difference between revisions of "2014 AMC 12B Problems"
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+ | {{AMC12 Problems|year=2014|ab=B}} | ||
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==Problem 1== | ==Problem 1== | ||
Leah has <math> 13 </math> coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | Leah has <math> 13 </math> coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | ||
− | <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qquad\textbf{(E)}\ 41 </math> |
[[2014 AMC 12B Problems/Problem 1|Solution]] | [[2014 AMC 12B Problems/Problem 1|Solution]] | ||
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Orvin went to the store with just enough money to buy <math> 30 </math> balloons. When he arrived he discovered that the store had a special sale on balloons: buy <math> 1 </math> balloon at the regular price and get a second at <math> \frac{1}{3} </math> off the regular price. What is the greatest number of balloons Orvin could buy? | Orvin went to the store with just enough money to buy <math> 30 </math> balloons. When he arrived he discovered that the store had a special sale on balloons: buy <math> 1 </math> balloon at the regular price and get a second at <math> \frac{1}{3} </math> off the regular price. What is the greatest number of balloons Orvin could buy? | ||
− | <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 39 </math> |
[[2014 AMC 12B Problems/Problem 2|Solution]] | [[2014 AMC 12B Problems/Problem 2|Solution]] | ||
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Randy drove the first third of his trip on a gravel road, the next <math> 20 </math> miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? | Randy drove the first third of his trip on a gravel road, the next <math> 20 </math> miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? | ||
− | <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ \frac{400}{11}\qquad\textbf{(C)}\ \frac{75}{2}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ \frac{400}{11}\qquad\textbf{(C)}\ \frac{75}{2}\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ \frac{300}{7} </math> |
[[2014 AMC 12B Problems/Problem 3|Solution]] | [[2014 AMC 12B Problems/Problem 3|Solution]] | ||
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Susie pays for <math> 4 </math> muffins and <math> 3 </math> bananas. Calvin spends twice as much paying for <math> 2 </math> muffins and <math> 16 </math> bananas. A muffin is how many times as expensive as a banana? | Susie pays for <math> 4 </math> muffins and <math> 3 </math> bananas. Calvin spends twice as much paying for <math> 2 </math> muffins and <math> 16 </math> bananas. A muffin is how many times as expensive as a banana? | ||
− | <math> \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \frac{5}{3}\qquad\textbf{(C)}\ \frac{7}{4}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \frac{5}{3}\qquad\textbf{(C)}\ \frac{7}{4}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{13}{4} </math> |
[[2014 AMC 12B Problems/Problem 4|Solution]] | [[2014 AMC 12B Problems/Problem 4|Solution]] | ||
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fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); | fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); | ||
</asy> | </asy> | ||
− | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math> |
[[2014 AMC 12B Problems/Problem 5|Solution]] | [[2014 AMC 12B Problems/Problem 5|Solution]] | ||
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Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume <math>\frac{3}{4}</math> of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together? | Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume <math>\frac{3}{4}</math> of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together? | ||
− | <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50 </math> |
[[2014 AMC 12B Problems/Problem 6|Solution]] | [[2014 AMC 12B Problems/Problem 6|Solution]] | ||
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For how many positive integers <math>n</math> is <math>\frac{n}{30-n}</math> also a positive integer? | For how many positive integers <math>n</math> is <math>\frac{n}{30-n}</math> also a positive integer? | ||
− | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math> |
[[2014 AMC 12B Problems/Problem 7|Solution]] | [[2014 AMC 12B Problems/Problem 7|Solution]] | ||
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Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, <math>abc</math> miles was displayed on the odometer, where <math>abc</math> is a 3-digit number with <math>a \geq{1}</math> and <math>a+b+c \leq{7}</math>. At the end of the trip, the odometer showed <math>cba</math> miles. What is <math>a^2+b^2+c^2?</math>. | Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, <math>abc</math> miles was displayed on the odometer, where <math>abc</math> is a 3-digit number with <math>a \geq{1}</math> and <math>a+b+c \leq{7}</math>. At the end of the trip, the odometer showed <math>cba</math> miles. What is <math>a^2+b^2+c^2?</math>. | ||
− | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37\qquad\textbf{(E)}\ 41 </math> |
[[2014 AMC 12B Problems/Problem 10|Solution]] | [[2014 AMC 12B Problems/Problem 10|Solution]] | ||
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A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list? | A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list? | ||
− | <math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D)}\ 33\qquad\textbf{(E)}\ 35 </math> |
[[2014 AMC 12B Problems/Problem 11|Solution]] | [[2014 AMC 12B Problems/Problem 11|Solution]] | ||
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A set <math>S</math> consists of triangles whose sides have integer lengths less than 5, and no two elements of <math>S</math> are congruent or similar. What is the largest number of elements that <math>S</math> can have? | A set <math>S</math> consists of triangles whose sides have integer lengths less than 5, and no two elements of <math>S</math> are congruent or similar. What is the largest number of elements that <math>S</math> can have? | ||
− | <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 </math> |
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==Problem 13== | ==Problem 13== | ||
− | Real numbers <math>a</math> and <math>b</math> are chosen with <math>1<a<b</math> such that no triangles with positive area has side lengths <math>1 | + | Real numbers <math>a</math> and <math>b</math> are chosen with <math>1<a<b</math> such that no triangles with positive area has side lengths <math>1, a,</math> and <math>b</math> or <math>\tfrac{1}{b}, \tfrac{1}{a},</math> and <math>1</math>. What is the smallest possible value of <math>b</math>? |
− | <math> \textbf{(A)}\ \frac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \frac{5}{2}\qquad\textbf{(C)}\ \frac{3+\sqrt{5}}{2}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ \frac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \frac{5}{2}\qquad\textbf{(C)}\ \frac{3+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 </math> |
[[2014 AMC 12B Problems/Problem 13|Solution]] | [[2014 AMC 12B Problems/Problem 13|Solution]] | ||
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A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? | A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? | ||
− | <math> \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2} </math> |
[[2014 AMC 12B Problems/Problem 14|Solution]] | [[2014 AMC 12B Problems/Problem 14|Solution]] | ||
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When <math>p = \sum\limits_{k=1}^{6} k \ln{k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math> ? | When <math>p = \sum\limits_{k=1}^{6} k \ln{k}</math>, the number <math>e^p</math> is an integer. What is the largest power of 2 that is a factor of <math>e^p</math> ? | ||
− | <math> \textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 2^{12}\qquad\textbf{(B)}\ 2^{14}\qquad\textbf{(C)}\ 2^{16}\qquad\textbf{(D)}\ 2^{18}\qquad\textbf{(E)}\ 2^{20} </math> |
[[2014 AMC 12B Problems/Problem 15|Solution]] | [[2014 AMC 12B Problems/Problem 15|Solution]] | ||
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Let <math>P</math> be a cubic polynomial with <math>P(0) = k</math>, <math>P(1) = 2k</math>, and <math>P(-1) = 3k</math>. What is <math>P(2) + P(-2)</math> ? | Let <math>P</math> be a cubic polynomial with <math>P(0) = k</math>, <math>P(1) = 2k</math>, and <math>P(-1) = 3k</math>. What is <math>P(2) + P(-2)</math> ? | ||
− | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ k\qquad\textbf{(C)}\ 6k\qquad\textbf{(D)}\ 7k\qquad\textbf{(E)}\ 14k </math> |
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Let <math>P</math> be the parabola with equation <math>y=x^2</math> and let <math>Q = (20, 14)</math>. There are real numbers <math>r</math> and <math>s</math> such that the line through <math>Q</math> with slope <math>m</math> does not intersect <math>P</math> if and only if <math>r < m < s</math>. What is <math>r + s</math>? | Let <math>P</math> be the parabola with equation <math>y=x^2</math> and let <math>Q = (20, 14)</math>. There are real numbers <math>r</math> and <math>s</math> such that the line through <math>Q</math> with slope <math>m</math> does not intersect <math>P</math> if and only if <math>r < m < s</math>. What is <math>r + s</math>? | ||
− | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 80 </math> |
[[2014 AMC 12B Problems/Problem 17|Solution]] | [[2014 AMC 12B Problems/Problem 17|Solution]] | ||
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==Problem 18== | ==Problem 18== | ||
− | The numbers 1, 2, 3, 4, 5, are to be arranged in a circle. An arrangement is <math>\textit{bad}</math> if it is not true that for every <math>n</math> from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? | + | The numbers <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, are to be arranged in a circle. An arrangement is <math>\textit{bad}</math> if it is not true that for every <math>n</math> from <math>1</math> to <math>15</math> one can find a subset of the numbers that appear consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? |
+ | |||
+ | <math> \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5 </math> | ||
[[2014 AMC 10B Problems/Problem_24|Solution]] | [[2014 AMC 10B Problems/Problem_24|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | + | A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? | |
+ | <asy> | ||
+ | real r=(3+sqrt(5))/2; | ||
+ | real s=sqrt(r); | ||
+ | real Brad=r; | ||
+ | real brad=1; | ||
+ | real Fht = 2*s; | ||
+ | import graph3; | ||
+ | import solids; | ||
+ | currentprojection=orthographic(1,0,.2); | ||
+ | currentlight=(10,10,5); | ||
+ | revolution sph=sphere((0,0,Fht/2),Fht/2); | ||
+ | //draw(surface(sph),green+white+opacity(0.5)); | ||
+ | //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} | ||
+ | triple f(pair t) { | ||
+ | triple v0 = Brad*(cos(t.x),sin(t.x),0); | ||
+ | triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); | ||
+ | return (v0 + t.y*(v1-v0)); | ||
+ | } | ||
+ | triple g(pair t) { | ||
+ | return (t.y*cos(t.x),t.y*sin(t.x),0); | ||
+ | } | ||
+ | surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); | ||
+ | surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); | ||
+ | surface base = surface(g,(0,0),(2pi,Brad),80,2); | ||
+ | draw(sback,gray(0.9)); | ||
+ | draw(sfront,gray(0.5)); | ||
+ | draw(base,gray(0.9)); | ||
+ | draw(surface(sph),gray(0.4));</asy> | ||
+ | <math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \quad \text{(D) } 2 \quad \text{(E) } \dfrac{3+\sqrt5}2</math> | ||
[[2014 AMC 12B Problems/Problem 19|Solution]] | [[2014 AMC 12B Problems/Problem 19|Solution]] | ||
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\textbf{(C) }19\qquad | \textbf{(C) }19\qquad | ||
\textbf{(D) }20\qquad | \textbf{(D) }20\qquad | ||
− | \textbf{(E) } | + | \textbf{(E) }\text{infinitely many}\qquad</math> |
[[2014 AMC 12B Problems/Problem 20|Solution]] | [[2014 AMC 12B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
+ | |||
+ | In the figure, <math> ABCD </math> is a square of side length <math> 1 </math>. The rectangles <math> JKHG </math> and <math> EBCF </math> are congruent. What is <math> BE </math>? | ||
+ | <asy> | ||
+ | pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(K--H--G--J--cycle); | ||
+ | draw(F--E); | ||
+ | label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); | ||
+ | label("$G$",G,E); label("$H$",H,N); label("$J$",J,S); label("$K$",K,W); | ||
+ | </asy> | ||
+ | <math> \textbf{(A) }\frac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\frac{\sqrt{3}}{6}\qquad\textbf{(E) } 1-\frac{\sqrt{2}}{2}</math> | ||
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==Problem 25== | ==Problem 25== | ||
− | Find the sum of all the positive solutions of | + | Find the sum of all the positive solutions of <cmath>2\cos(2x) \left(\cos(2x) - \cos\left( \frac{2014\pi^2}{x} \right)\right) = \cos(4x) - 1</cmath> |
− | <math> | + | <math> \textbf{(A)}\ \pi \qquad\textbf{(B)}\ 810\pi \qquad\textbf{(C)}\ 1008\pi \qquad\textbf{(D)}\ 1080 \pi \qquad\textbf{(E)}\ 1800\pi </math> |
− | + | [[2014 AMC 12B Problems/Problem 25|Solution]] | |
− | [[2014 AMC | + | ==See also== |
+ | |||
+ | {{AMC12 box|year=2014|ab=B|before=[[2014 AMC 12A Problems]]|after=[[2015 AMC 12A Problems]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 14:08, 1 November 2022
2014 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
Leah has coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
Problem 2
Orvin went to the store with just enough money to buy balloons. When he arrived he discovered that the store had a special sale on balloons: buy
balloon at the regular price and get a second at
off the regular price. What is the greatest number of balloons Orvin could buy?
Problem 3
Randy drove the first third of his trip on a gravel road, the next miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
Problem 4
Susie pays for muffins and
bananas. Calvin spends twice as much paying for
muffins and
bananas. A muffin is how many times as expensive as a banana?
Problem 5
Doug constructs a square window using equal-size panes of glass, as shown. The ratio of the height to width for each pane is
, and the borders around and between the panes are
inches wide. In inches, what is the side length of the square window?
Problem 6
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Problem 7
For how many positive integers is
also a positive integer?
Problem 8
In the addition shown below ,
,
, and
are distinct digits. How many different values are possible for
?
Problem 9
Convex quadrilateral has
,
,
,
, and
, as shown. What is the area of the quadrilateral?
Problem 10
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, miles was displayed on the odometer, where
is a 3-digit number with
and
. At the end of the trip, the odometer showed
miles. What is
.
Problem 11
A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?
Problem 12
A set consists of triangles whose sides have integer lengths less than 5, and no two elements of
are congruent or similar. What is the largest number of elements that
can have?
Problem 13
Real numbers and
are chosen with
such that no triangles with positive area has side lengths
and
or
and
. What is the smallest possible value of
?
Problem 14
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
Problem 15
When , the number
is an integer. What is the largest power of 2 that is a factor of
?
Problem 16
Let be a cubic polynomial with
,
, and
. What is
?
Problem 17
Let be the parabola with equation
and let
. There are real numbers
and
such that the line through
with slope
does not intersect
if and only if
. What is
?
Problem 18
The numbers ,
,
,
,
, are to be arranged in a circle. An arrangement is
if it is not true that for every
from
to
one can find a subset of the numbers that appear consecutively on the circle that sum to
. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
Problem 19
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
Problem 20
For how many positive integers is
?
Problem 21
In the figure, is a square of side length
. The rectangles
and
are congruent. What is
?
Problem 22
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad ,
, it will jump to pad
with probability
and to pad
with probability
. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?
Problem 23
The number 2017 is prime. Let . What is the remainder when
is divided by 2017?
Problem 24
Let be a pentagon inscribed in a circle such that
,
, and
. The sum of the lengths of all diagonals of
is equal to
, where
and
are relatively prime positive integers. What is
?
Problem 25
Find the sum of all the positive solutions of
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2014 AMC 12A Problems |
Followed by 2015 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.