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Difference between revisions of "2014 AMC 10A Problems"
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==duck dynasty==
+
{{AMC10 Problems|year=2014|ab=A}}
 +
==Problem 1==
  
 
What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math>
 
What is <math>10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?</math>
  
−
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ \frac{25}{2}\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math>
+
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170</math>
  
 
[[2014 AMC 10A  Problems/Problem 1|Solution]]
 
[[2014 AMC 10A  Problems/Problem 1|Solution]]
Line 9: Line 10:
 
==Problem 2==
 
==Problem 2==
  
−
Roy's rat eats <math>\frac{1}{3}</math> of a can of bat food every morning and <math>\frac{1}{4}</math> of a can of fat food every evening. Before feeding his gnat on Monday morning, Roy opened a box containing <math>6</math> cans of scat food. On what day of the week did the hat finish eating all the mat food in the box?
+
Roy's cat eats <math>\frac{1}{3}</math> of a can of cat food every morning and <math>\frac{1}{4}</math> of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing <math>6</math> cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
  
−
<math> \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}\qquad\textbf{(F)}\ \text{Option 6}</math>
+
<math> \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}</math>
  
−
[[2014 AMC 10A  Problems/Problem 2|onion]]
+
[[2014 AMC 10A  Problems/Problem 2|Solution]]
 +
==Problem 3==
  
−
==Half-Life 3==
+
Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for <math>\textdollar 2.50</math> each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs <math>\textdollar 0.75</math> for her to make. In dollars, what is her profit for the day?
−
 
 
−
Jeb Bush bakes 48 billionaires for her campaign. Please clap. She sells half of them in the morning for <math>\textdollar 2.50</math> each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining billionaires at a dollar each. Each billionaire costs <math>\textdollar 0.75</math> for her to make. In dollars, what is her profit for the day? #nevertrump #jeb
 
  
 
<math>\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52</math>
 
<math>\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52</math>
  
 
[[2014 AMC 10A  Problems/Problem 3|Solution]]
 
[[2014 AMC 10A  Problems/Problem 3|Solution]]
−
 
 
==Problem 4==
 
==Problem 4==
  
−
Ralphing down Ralph Street, Ralph passed four Ralphs in a row, each Ralphed a different color. He Ralphed the Ralph house before the red Ralph, and he passed the Ralph Ralph before the county jail. The Ralph Ralph was not next to the county jail. How many orderings of the colored Ralphs are possible?
+
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
  
−
<math>\textbf{(A)}\ 2\qquad\textbf{(ok boomer)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ \frac{1}{\frac{2}{\frac{3}{\frac{4}{5}}}}\qquad\textbf{(F)}\ in  the chat </math>
+
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
  
 
[[2014 AMC 10A  Problems/Problem 4|Solution]]
 
[[2014 AMC 10A  Problems/Problem 4|Solution]]
 +
==Problem 5==
  
−
==Problematic Grades==
+
On an algebra quiz, <math>10\%</math> of the students scored <math>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</math> points. What is the difference between the mean and median score of the students' scores on this quiz?
−
 
 
−
On an algebra quiz, <math>10\%</math> of the students scored <math>0</math> points, <math>35\%</math> scored <math>10</math> points, <math>30\%</math> scored <math>15</math> points, and the rest scored <math>30</math> points. What is the difference between the mean and median score of the students' scores on this quiz?
 
  
 
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math>
 
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math>
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==Problem 6==
 
==Problem 6==
  
−
Suppose that you are a cow.
+
Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days?
  
−
<math>\ \frac{bcde}{a}=109\ gallons</math>
+
<math> \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}</math>
  
 
[[2014 AMC 10A  Problems/Problem 6|Solution]]
 
[[2014 AMC 10A  Problems/Problem 6|Solution]]
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==Problem 7==
 
==Problem 7==
  
−
Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true?  
+
Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true?
  
 
<math>\textbf{(I)}\ x+y < a+b\qquad</math>
 
<math>\textbf{(I)}\ x+y < a+b\qquad</math>
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[[2014 AMC 10A  Problems/Problem 9|Solution]]
 
[[2014 AMC 10A  Problems/Problem 9|Solution]]
−
 
 
==Problem 10==
 
==Problem 10==
  
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For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>?
 
For which of the following listed prices will coupon <math>1</math> offer a greater price reduction than either coupon <math>2</math> or coupon <math>3</math>?
 +
 
<math>\textbf{(A) }\textdollar179.95\qquad
 
<math>\textbf{(A) }\textdollar179.95\qquad
 
\textbf{(B) }\textdollar199.95\qquad
 
\textbf{(B) }\textdollar199.95\qquad
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[[2014 AMC 10A  Problems/Problem 11|Solution]]
 
[[2014 AMC 10A  Problems/Problem 11|Solution]]
−
 
 
==Problem 12==
 
==Problem 12==
  
Line 117: Line 114:
 
defaultpen(linewidth(0.8));
 
defaultpen(linewidth(0.8));
 
path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle;
 
path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle;
−
fill(hexagon,yellow);
+
fill(hexagon,lightgrey);
−
for(int i=0;i<=27;i=2i+1)
+
for(int i=0;i<=5;i=i+1)
 
{
 
{
−
path arc=2*dir(63*i)--arc(2*dir(62*i),1,120+61*i,240+59*i)--cycle;
+
path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle;
 
unfill(arc);
 
unfill(arc);
 
draw(arc);
 
draw(arc);
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Equilateral <math>\triangle ABC</math> has side length <math>1</math>, and squares <math>ABDE</math>, <math>BCHI</math>, <math>CAFG</math> lie outside the triangle. What is the area of hexagon <math>DEFGHI</math>?
 
Equilateral <math>\triangle ABC</math> has side length <math>1</math>, and squares <math>ABDE</math>, <math>BCHI</math>, <math>CAFG</math> lie outside the triangle. What is the area of hexagon <math>DEFGHI</math>?
 +
 
<asy>
 
<asy>
 
import graph;
 
import graph;
 
size(6cm);
 
size(6cm);
−
pen dps = linewidth(0.7) + fontsize(11); defaultpen(dps);
+
pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps);
 
pair B = (0,0);
 
pair B = (0,0);
 
pair C = (1,0);
 
pair C = (1,0);
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pair E = rotate(270,A)*B;
 
pair E = rotate(270,A)*B;
−
pair D = rotate(275,E)*A;
+
pair D = rotate(270,E)*A;
  
−
pair F = rotate(92,A)*C;
+
pair F = rotate(90,A)*C;
 
pair G = rotate(90,F)*A;
 
pair G = rotate(90,F)*A;
  
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draw(A--B--C--cycle);
 
draw(A--B--C--cycle);
−
draw(A--E--C--B);
+
draw(A--E--D--B);
 
draw(A--F--G--C);
 
draw(A--F--G--C);
 
draw(B--I--H--C);
 
draw(B--I--H--C);
  
 
draw(E--F);
 
draw(E--F);
−
draw(F--I);
+
draw(D--I);
 
draw(I--H);
 
draw(I--H);
 
draw(H--G);
 
draw(H--G);
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[[2014 AMC 10A  Problems/Problem 13|Solution]]
 
[[2014 AMC 10A  Problems/Problem 13|Solution]]
−
 
 
==Problem 14==
 
==Problem 14==
  
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==Problem 15==
 
==Problem 15==
  
−
Number 15: Burger king foot lettuce. The last thing you'd want in your Burger King burger is someone's foot fungus. But as it turns out, that might be what you get. A 4channer uploaded a photo anonymously to the site showcasing his feet in a plastic bin of lettuce. With the statement: "This is the lettuce you eat at Burger King." Admittedly, he had shoes on.
+
David drives from his home to the airport to catch a flight. He drives <math>35</math> miles in the first hour, but realizes that he will be <math>1</math> hour late if he continues at this speed. He increases his speed by <math>15</math> miles per hour for the rest of the way to the airport and arrives <math>30</math> minutes early. How many miles is the airport from his home?
−
 
 
−
But that's even worse.
 
−
 
 
−
How many miles is the airport from his home?
 
  
 
<math>\textbf{(A) }140\qquad
 
<math>\textbf{(A) }140\qquad
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[[2014 AMC 10A  Problems/Problem 15|Solution]]
 
[[2014 AMC 10A  Problems/Problem 15|Solution]]
−
 
 
==Problem 16==
 
==Problem 16==
  
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size(9cm);
 
size(9cm);
 
pen dps = fontsize(10); defaultpen(dps);
 
pen dps = fontsize(10); defaultpen(dps);
−
pair D = (0,1/5);
+
pair D = (0,0);
−
pair F = (1/2,2/3);
+
pair F = (1/2,0);
−
pair C = (1,2);
+
pair C = (1,0);
−
pair G = (3,1);
+
pair G = (0,1);
−
pair E = (1,2);
+
pair E = (1,1);
 
pair A = (0,2);
 
pair A = (0,2);
 
pair B = (1,2);
 
pair B = (1,2);
−
pair H = (1/3,4);
+
pair H = (1/2,1);
  
 
// do not look
 
// do not look
−
pair X = (1/7,2/3);
+
pair X = (1/3,2/3);
−
pair Y = (2/3,2/9);
+
pair Y = (2/3,2/3);
  
−
draw(A--G--C--D--cycle);
+
draw(A--B--C--D--cycle);
 
draw(G--E);
 
draw(G--E);
−
draw(A--C--B);
+
draw(A--F--B);
−
draw(D--E--C);
+
draw(D--H--C);
−
filldraw(H--A--F--B--cycle,grey);
+
filldraw(H--X--F--Y--cycle,grey);
  
−
label("$A$",B,NW);
+
label("$A$",A,NW);
 
label("$B$",B,NE);
 
label("$B$",B,NE);
−
label("$C$",B,SE);
+
label("$C$",C,SE);
−
label("$D$",B,SW);
+
label("$D$",D,SW);
−
label("$E$",B,E);
+
label("$E$",E,E);
−
label("$F$",B,S);
+
label("$F$",F,S);
−
label("$G$",B,W);
+
label("$G$",G,W);
−
label("$H$",B,N);
+
label("$H$",H,N);
  
 
label("$\frac12$",(0.25,0),S);
 
label("$\frac12$",(0.25,0),S);
 
label("$\frac12$",(0.75,0),S);
 
label("$\frac12$",(0.75,0),S);
−
label("$1$",(2,0.5),E);
+
label("$1$",(1,0.5),E);
 
label("$1$",(1,1.5),E);
 
label("$1$",(1,1.5),E);
 
</asy>
 
</asy>
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==Problem 17==
 
==Problem 17==
  
−
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?‮
+
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
  
 
<math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math>
 
<math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math>
  
 
[[2014 AMC 10A  Problems/Problem 17|Solution]]
 
[[2014 AMC 10A  Problems/Problem 17|Solution]]
−
 
 
==Problem 18==
 
==Problem 18==
  
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label("$2$", (2,8), W,fontsize(8pt));
 
label("$2$", (2,8), W,fontsize(8pt));
 
label("$3$", (3,5.5), W,fontsize(8pt));
 
label("$3$", (3,5.5), W,fontsize(8pt));
−
label("$make america great again$", (4,2), W,fontsize(8pt));
+
label("$4$", (4,2), W,fontsize(8pt));
 
</asy>
 
</asy>
 
[[2014 AMC 10A  Problems/Problem 19|Solution]]
 
[[2014 AMC 10A  Problems/Problem 19|Solution]]
−
 
 
==Problem 20==
 
==Problem 20==
  
−
The product <math>(8)(888\dots8)</math>, where the second factor has <math>991</math> digits, is an integer whose digits have a sum of <math>1000</math>.  What is <math>991</math>?
+
The product <math>(8)(888\dots8)</math>, where the second factor has <math>k</math> digits, is an integer whose digits have a sum of <math>1000</math>.  What is <math>k</math>?
  
−
<math>\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999</math>
+
<math> \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999 </math>
  
 
[[2014 AMC 10A  Problems/Problem 20|Solution]]
 
[[2014 AMC 10A  Problems/Problem 20|Solution]]
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==Problem 24==
 
==Problem 24==
  
−
A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000</math>th number in the sequence?
+
A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000^{\text{th}}</math> number in the sequence?
  
 
<math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math>
 
<math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math>
Line 353: Line 343:
 
[[2014 AMC 10A  Problems/Problem 24|Solution]]
 
[[2014 AMC 10A  Problems/Problem 24|Solution]]
  
−
== Problem 25==
+
==Problem 25==
  
 
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>.  How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath>
 
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>.  How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath>
Line 360: Line 350:
  
 
[[2014 AMC 10A  Problems/Problem 25|Solution]]
 
[[2014 AMC 10A  Problems/Problem 25|Solution]]
−
 
 
==See also==
 
==See also==
 
{{AMC10 box|year=2014|ab=A|before=[[2013 AMC 10B Problems]]|after=[[2014 AMC 10B Problems]]}}
 
{{AMC10 box|year=2014|ab=A|before=[[2013 AMC 10B Problems]]|after=[[2014 AMC 10B Problems]]}}

Latest revision as of 15:54, 28 July 2024

2014 AMC 10A (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

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

Solution

Problem 2

Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?

$\textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Solution

Problem 3

Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\textdollar 2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\textdollar 0.75$ for her to make. In dollars, what is her profit for the day?

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52$

Solution

Problem 4

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

Problem 5

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?

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

Solution

Problem 6

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days?

$\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

Solution

Problem 7

Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?

$\textbf{(I)}\ x+y < a+b\qquad$

$\textbf{(II)}\ x-y < a-b\qquad$

$\textbf{(III)}\ xy < ab\qquad$

$\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution

Problem 8

Which of the following numbers is a perfect square?

$\textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2$

Solution

Problem 9

The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle?

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

Solution

Problem 10

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?

$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7$

Solution

Problem 11

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: $10\%$ off the listed price if the listed price is at least $\textdollar50$

Coupon 2: $\textdollar 20$ off the listed price if the listed price is at least $\textdollar100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $\textdollar100$

For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?

$\textbf{(A) }\textdollar179.95\qquad \textbf{(B) }\textdollar199.95\qquad \textbf{(C) }\textdollar219.95\qquad \textbf{(D) }\textdollar239.95\qquad \textbf{(E) }\textdollar259.95\qquad$

Solution

Problem 12

A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?

[asy] size(125); defaultpen(linewidth(0.8)); path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle; fill(hexagon,lightgrey); for(int i=0;i<=5;i=i+1) { path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle; unfill(arc); draw(arc); } draw(hexagon,linewidth(1.8));[/asy]

$\textbf{(A)}\ 27\sqrt{3}-9\pi\qquad\textbf{(B)}\ 27\sqrt{3}-6\pi\qquad\textbf{(C)}\ 54\sqrt{3}-18\pi\qquad\textbf{(D)}\ 54\sqrt{3}-12\pi\qquad\textbf{(E)}\ 108\sqrt{3}-9\pi$

Solution

Problem 13

Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,W); label("$E$",E,W); label("$F$",F,E); label("$G$",G,E); label("$H$",H,SE); label("$I$",I,SW); [/asy]

$\textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6$

Solution

Problem 14

The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?

$\textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72$

Solution

Problem 15

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?

$\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

Solution

Problem 16

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?

[asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\frac12$",(0.25,0),S); label("$\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy]

$\textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16$

Solution

Problem 17

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

$\textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29$

Solution

Problem 18

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?

$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27$

Solution

Problem 19

Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$?

$\textbf{(A)}\ \dfrac{3\sqrt{33}}5\qquad\textbf{(B)}\ 2\sqrt3\qquad\textbf{(C)}\ \dfrac{2\sqrt{33}}3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 3\sqrt2$

[asy] dotfactor = 3; size(10cm); dot((0, 10)); label("$X$", (0,10),W,fontsize(8pt)); dot((6,2)); label("$Y$", (6,2),E,fontsize(8pt)); draw((0, 0)--(0, 10)--(1, 10)--(1, 9)--(2, 9)--(2, 7)--(3, 7)--(3,4)--(4, 4)--(4, 0)--cycle); draw((0,9)--(1, 9)--(1.5, 9.5)--(1.5, 10.5)--(0.5, 10.5)--(0, 10)); draw((1, 10)--(1.5,10.5)); draw((1.5, 10)--(3,10)--(3,8)--(2,7)--(0,7)); draw((2,9)--(3,10)); draw((3,8.5)--(4.5,8.5)--(4.5,5.5)--(3,4)--(0,4)); draw((3,7)--(4.5,8.5)); draw((4.5,6)--(6,6)--(6,2)--(4,0)); draw((4,4)--(6,6)); label("$1$", (1,9.5), W,fontsize(8pt)); label("$2$", (2,8), W,fontsize(8pt)); label("$3$", (3,5.5), W,fontsize(8pt)); label("$4$", (4,2), W,fontsize(8pt)); [/asy] Solution

Problem 20

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?

$\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}\ 991\qquad\textbf{(E)}\ 999$

Solution

Problem 21

Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?

$\textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8}$

Solution

Problem 22

In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?

$\textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20$

Solution

Problem 23

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?

[asy] import graph; size(6cm); real L = 0.05; pair A = (0,0); pair B = (sqrt(3),0); pair C = (sqrt(3),1); pair D = (0,1); pair X1 = (sqrt(3)/3,0); pair X2= (2*sqrt(3)/3,0); pair Y1 = (2*sqrt(3)/3,1); pair Y2 = (sqrt(3)/3,1); dot(X1); dot(Y1); draw(A--B--C--D--cycle, linewidth(2)); draw(X1--Y1,dashed); draw(X2--(2*sqrt(3)/3,L)); draw(Y2--(sqrt(3)/3,1-L)); [/asy]

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

Solution

Problem 24

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000^{\text{th}}$ number in the sequence?

$\textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510$

Solution

Problem 25

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]

$\textbf{(A) }278\qquad\textbf{(B) }279\qquad\textbf{(C) }280\qquad\textbf{(D) }281\qquad\textbf{(E) }282\qquad$

Solution

See also

2014 AMC 10A (Problems • Answer Key • Resources)
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2013 AMC 10B Problems 		Followed by
2014 AMC 10B Problems
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All AMC 10 Problems and Solutions

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