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Difference between revisions of "2014 AMC 10B Problems"

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{{AMC10 Problems|year=2014|ab=B}}
 
==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 as 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) } 39 \qquad \textbf {(E) } 41 </math>
 
<math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41 </math>
 +
 +
[[2014 AMC 10B Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
 
What is <math>\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}</math>?
 
What is <math>\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}</math>?
  
−
<math> \textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64 </math>
+
<math>\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64</math>
 +
 
 +
[[2014 AMC 10B Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
  
−
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?
+
Peter 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 Peter's trip?
  
 
<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>
 
<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 10B Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
Line 21: Line 28:
 
<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>
 
<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 10B Problems/Problem 4|Solution]]
 
==Problem 5==
 
==Problem 5==
 +
 +
Camden constructs a square window using <math> 8 </math> equal-size panes of glass, as shown. The ratio of the height to width for each pane is <math> 5 : 2 </math>, and the borders around and between the panes are <math> 2 </math> inches wide. In inches, what is the side length of the square window?
 +
<asy>
 +
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
 +
for(int i = 0; i < 4; ++i){
 +
  for(int j = 0; j < 2; ++j){
 +
    fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
 +
  }
 +
}</asy>
 +
<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math>
 +
 +
[[2014 AMC 10B Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
Line 29: Line 49:
 
<math> \textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39</math>
 
<math> \textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39</math>
  
 +
[[2014 AMC 10B Problems/Problem 6|Solution]]
 
==Problem 7==
 
==Problem 7==
  
−
Suppose <math>A>B>0</math> and A is <math>x%</math> greater than <math>B</math>. What is <math>x</math>?
+
Suppose <math>A>B>0</math> and <math>A</math> is <math>x\%</math> greater than <math>B</math>. What is <math>x</math>?
 +
 
 +
<math> \textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)</math>
  
−
<math> \textbf {(A) } 100(\frac{A-B}{B}) \qquad \textbf {(B) } 100(\frac{A+B}{B}) \qquad \textbf {(C) } 100(\frac{A+B}{A})\qquad \textbf {(D) } 100(\frac{A-B}{A}) \qquad \textbf {(E) } 100(\frac{A}{B})</math>
+
[[2014 AMC 10B Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
−
A truck travels <math>{b}{6}</math> feet ever <math>t</math> seconds. There are <math>3</math> feet in a yard. How many yards does the truck travel in <math>3</math> minutes?
+
A truck travels <math>\frac{b}{6}</math> feet every <math>t</math> seconds. There are <math>3</math> feet in a yard. How many yards does the truck travel in <math>3</math> minutes?
  
 
<math> \textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}</math>
 
<math> \textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}</math>
  
 +
[[2014 AMC 10B Problems/Problem 8|Solution]]
 
==Problem 9==
 
==Problem 9==
 +
For real numbers <math>w</math> and <math>z</math>,
 +
<cmath>\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.</cmath>
 +
What is <math>\frac{w+z}{w-z}</math>?
 +
 +
<math>\textbf{(A) } -2014 \qquad\textbf{(B) } \frac{-1}{2014} \qquad\textbf{(C) } \frac{1}{2014} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2014</math>
 +
 +
[[2014 AMC 10B Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
 +
In the addition shown below <math>A, B, C,</math> and <math>D</math> are distinct digits. How many different values are possible for <math>D</math>?
 +
 +
<cmath> \begin{array}{lr}&ABBCB\\ +& BCADA\\ \hline & DBDDD\end{array} </cmath>
 +
 +
<math> \textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9 </math>
 +
 +
[[2014 AMC 10B Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
For the consumer, a single discount of <math>n\%</math> is more advantageous than any of the following discounts:
 +
 +
(1) two successive <math>15\%</math>  discounts
 +
 +
(2) three successive <math>10\%</math> discounts
 +
 +
(3) a <math>25\%</math> discount followed by a <math>5\%</math> discount
 +
 +
What is the smallest possible positive integer value of <math>n</math>?
 +
 +
<math> \textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33 </math>
 +
 +
[[2014 AMC 10B Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth largest divisor?
 +
 +
<math> \textbf{(A)}\ \ 125,875,000\qquad\textbf{(B)}\ 201,400,000\qquad\textbf{(C)}\ 251,750,000\qquad\textbf{(D)}\ 402,800,000\qquad\textbf{(E)}\ 503,500,000 </math>
 +
 +
[[2014 AMC 10B Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
Six regular hexagons surround a regular hexagon of side length <math>1</math> as shown. What is the area of <math>\triangle ABC</math>?
 +
 +
<asy>
 +
for(int i = 0; i < 6; ++i){
 +
for(int j = 0; j < 6; ++j){
 +
  draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60));
 +
}
 +
}
 +
 +
draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle);
 +
label("A",2*dir(180),dir(180));
 +
label("B",2*dir(60),dir(60));
 +
label("C",2*dir(300),dir(300));
 +
</asy>
 +
 +
 +
<math> \textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3} </math>
 +
 +
[[2014 AMC 10B Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
Danica drove her new car on a trip for a whole number of hours, averaging <math>55</math> 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 \ge 1</math> and <math>a + b + c \le 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) } 37 \qquad \textbf {(E) }41</math>
 +
 +
[[2014 AMC 10B Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
In rectangle <math>ABCD</math>, <math>DC=2 \cdot CB</math> and points <math>E</math> and <math>F</math> lie on <math>\overline{AB}</math> so that <math>\overline{ED}</math> and <math>\overline{FD}</math> trisect <math>\angle ADC</math> as shown. What is the ratio of the area of <math>\triangle DEF</math> to the area of rectangle <math>ABCD</math>?
 +
 +
<asy>
 +
pair A = (0,1), B = (2,1), C = (2,0), D = (0,0);
 +
pair E = intersectionpoint(A--B,D--2*dir(60)), F = intersectionpoint(A--B,D--3*dir(30));
 +
 +
draw(A--D--C--B--cycle);
 +
draw(E--D--F);
 +
label("A",A,N);
 +
label("B",B,N);
 +
label("C",C,S);
 +
label("D",D,S);
 +
label("E",E,N);
 +
label("F",F,N);
 +
</asy>
 +
 +
<math> \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4} </math>
 +
 +
[[2014 AMC 10B Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
  
 +
<math> \textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9}\qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}</math>
 +
 +
[[2014 AMC 10B Problems/Problem 16|Solution]]
 
==Problem 17==
 
==Problem 17==
 +
What is the greatest power of <math>2</math> that is a factor of <math>10^{1002} - 4^{501}</math>?
 +
 +
<math>\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}</math>
  
 +
[[2014 AMC 10B Problems/Problem 17|Solution]]
 
==Problem 18==
 
==Problem 18==
 +
A list of <math>11</math> positive integers has a mean of <math>10</math>, a median of <math>9</math>, and a unique mode of <math>8</math>. 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) } 33 \qquad \textbf {(E) } 35</math>
  
 +
[[2014 AMC 10B Problems/Problem 18|Solution]]
 
==Problem 19==
 
==Problem 19==
 +
Two concentric circles have radii <math>1</math> and <math>2</math>. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
  
 +
<math>\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad</math>
 +
 +
[[2014 AMC 10B Problems/Problem 19|Solution]]
 
==Problem 20==
 
==Problem 20==
 +
For how many integers <math>x</math> is the number <math>x^4 - 51x^2 + 50</math> negative?
  
 +
<math> \textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12\qquad \textbf {(D) } 14 \qquad \textbf {(E) }16</math>
 +
 +
[[2014 AMC 10B Problems/Problem 20|Solution]]
 
==Problem 21==
 
==Problem 21==
 +
Trapezoid <math>ABCD</math> has parallel sides <math>\overline{AB}</math> of length <math>33</math> and <math>\overline{CD}</math> of length <math>21</math>. The other two sides are of lengths <math>10</math> and <math>14</math>. The angles at <math>A</math> and <math>B</math> are acute. What is the length of the shorter diagonal of <math>ABCD</math>?
  
 +
<math> \textbf{(A) } 10\sqrt{6} \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 8\sqrt{10} \qquad\textbf{(D) } 18\sqrt{2} \qquad\textbf{(E) } 26</math>
 +
 +
[[2014 AMC 10B Problems/Problem 21|Solution]]
 
==Problem 22==
 
==Problem 22==
 +
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
 +
 +
<math>\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3</math>
 +
 +
<asy>
 +
scale(200);
 +
draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle));
 +
path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180);
 +
draw(p);
 +
p=rotate(90)*p; draw(p);
 +
p=rotate(90)*p; draw(p);
 +
p=rotate(90)*p; draw(p);
 +
draw(scale((sqrt(5)-1)/4)*unitcircle);
 +
</asy>
 +
 +
[[2014 AMC 10B Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
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(0.3));
 +
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 10B Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is ''bad'' 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]]
 
==Problem 25==
 
==Problem 25==
 +
In a small pond there are eleven lily pads in a row labeled <math>0</math> through <math>10</math>. A frog is sitting on pad <math>1</math>. When the frog is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</math> with probability <math>1-\frac{N}{10}</math>. Each jump is independent of the previous jumps. If the frog reaches pad <math>0</math> it will be eaten by a patiently waiting snake. If the frog reaches pad <math>10</math> it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?
 +
 +
<math> \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} </math>
 +
 +
[[2014 AMC 10B Problems/Problem 25|Solution]]
 +
 +
== See also ==
 +
{{AMC10 box|year=2014|ab=B|before=[[2014 AMC 10A Problems]]|after=[[2015 AMC 10A Problems]]}}
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[2014 AMC 10B]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 20:54, 8 November 2022

2014 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

Leah has $13$ 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 as nickels. In cents, how much are Leah's coins worth?

$\textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 41$

Solution

Problem 2

What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$?

$\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D) } 48 \qquad \textbf {(E) } 64$

Solution

Problem 3

Peter drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip?

$\textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7}$

Solution

Problem 4

Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?

$\textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$

Solution

Problem 5

Camden constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? [asy] fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey); for(int i = 0; i < 4; ++i){ for(int j = 0; j < 2; ++j){ fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white); } }[/asy] $\textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34$

Solution

Problem 6

Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?

$\textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39$

Solution

Problem 7

Suppose $A>B>0$ and $A$ is $x\%$ greater than $B$. What is $x$?

$\textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)$

Solution

Problem 8

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?

$\textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$

Solution

Problem 9

For real numbers $w$ and $z$, \[\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.\] What is $\frac{w+z}{w-z}$?

$\textbf{(A) } -2014 \qquad\textbf{(B) } \frac{-1}{2014} \qquad\textbf{(C) } \frac{1}{2014} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2014$

Solution

Problem 10

In the addition shown below $A, B, C,$ and $D$ are distinct digits. How many different values are possible for $D$?

\[\begin{array}{lr}&ABBCB\\ +& BCADA\\ \hline & DBDDD\end{array}\]

$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Solution

Problem 11

For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:

(1) two successive $15\%$ discounts

(2) three successive $10\%$ discounts

(3) a $25\%$ discount followed by a $5\%$ discount

What is the smallest possible positive integer value of $n$?

$\textbf{(A)}\ \ 27\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33$

Solution

Problem 12

The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor?

$\textbf{(A)}\ \ 125,875,000\qquad\textbf{(B)}\ 201,400,000\qquad\textbf{(C)}\ 251,750,000\qquad\textbf{(D)}\ 402,800,000\qquad\textbf{(E)}\ 503,500,000$

Solution

Problem 13

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$?

[asy] for(int i = 0; i < 6; ++i){ for(int j = 0; j < 6; ++j){ draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60)); } } draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle); label("A",2*dir(180),dir(180)); label("B",2*dir(60),dir(60)); label("C",2*dir(300),dir(300)); [/asy]


$\textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3}$

Solution

Problem 14

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, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a + b + c \le 7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2 + b^2 + c^2$ ?

$\textbf {(A) } 26 \qquad \textbf {(B) } 27 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 37 \qquad \textbf {(E) }41$

Solution

Problem 15

In rectangle $ABCD$, $DC=2 \cdot CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?

[asy] pair A = (0,1), B = (2,1), C = (2,0), D = (0,0); pair E = intersectionpoint(A--B,D--2*dir(60)), F = intersectionpoint(A--B,D--3*dir(30)); draw(A--D--C--B--cycle); draw(E--D--F); label("A",A,N); label("B",B,N); label("C",C,S); label("D",D,S); label("E",E,N); label("F",F,N); [/asy]

$\textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$

Solution

Problem 16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

$\textbf {(A) } \frac{1}{36} \qquad \textbf {(B) } \frac{7}{72} \qquad \textbf {(C) } \frac{1}{9}\qquad \textbf {(D) } \frac{5}{36} \qquad \textbf {(E) } \frac{1}{6}$

Solution

Problem 17

What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?

$\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}$

Solution

Problem 18

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?

$\textbf {(A) } 24 \qquad \textbf {(B) } 30 \qquad \textbf {(C) } 31\qquad \textbf {(D) } 33 \qquad \textbf {(E) } 35$

Solution

Problem 19

Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$

Solution

Problem 20

For how many integers $x$ is the number $x^4 - 51x^2 + 50$ negative?

$\textbf {(A) } 8 \qquad \textbf {(B) } 10 \qquad \textbf {(C) } 12\qquad \textbf {(D) } 14 \qquad \textbf {(E) }16$

Solution

Problem 21

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$?

$\textbf{(A) } 10\sqrt{6} \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 8\sqrt{10} \qquad\textbf{(D) } 18\sqrt{2} \qquad\textbf{(E) } 26$

Solution

Problem 22

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

$\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3$

[asy] scale(200); draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle)); path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180); draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); p=rotate(90)*p; draw(p); draw(scale((sqrt(5)-1)/4)*unitcircle); [/asy]

Solution

Problem 23

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(0.3)); draw(sfront,gray(0.5)); draw(base,gray(0.9)); draw(surface(sph),gray(0.4));[/asy]

$\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \quad \text{(D) } 2 \quad \text{(E) } \dfrac{3+\sqrt5}2$

Solution

Problem 24

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

$\textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5$

Solution

Problem 25

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 $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. 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?

$\textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2}$

Solution

See also

2014 AMC 10B (Problems • Answer Key • Resources)
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2014 AMC 10A Problems 		Followed by
2015 AMC 10A Problems
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All AMC 10 Problems and Solutions

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