Difference between revisions of "2014 AMC 10A Problems"

(Undo revision 115472 by Happycupcake (talk))
(Tag: Undo)
(Undo revision 115476 by Arnold (talk))
(Tag: Undo)
Line 1: Line 1:
 
+
==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>
Line 6: Line 6:
  
 
[[2014 AMC 10A  Problems/Problem 1|Solution]]
 
[[2014 AMC 10A  Problems/Problem 1|Solution]]
 
==Arnold Murphy's Bologna dare==
 
  
 
==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 Roy's 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}\qquad\textbf{(F)}\ \text{Option 6}</math>
  
[[2014 AMC 10A  Problems/Problem 2|onion]]
+
[[2014 AMC 10A  Problems/Problem 2|solution]]
  
==Half-Life 3==
+
==Problem 3==
  
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
+
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?
  
 
<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>
Line 27: Line 25:
 
==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{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
 
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
Line 33: Line 31:
 
[[2014 AMC 10A  Problems/Problem 4|Solution]]
 
[[2014 AMC 10A  Problems/Problem 4|Solution]]
  
==Problematic Grades==
+
==Problem 5==
  
 
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?
 
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?
Line 43: Line 41:
 
==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]]
Line 64: Line 62:
 
\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math>
 
\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math>
  
<math> \textbf{(O)}\qquad\textbf{(K)}\qquad\textbf{(B)}\qquad\textbf{(O)}\qquad\textbf{(O)}\qquad\textbf{(M)}\qquad\textbf{(E)}\qquad\textbf{(R)}</math>
+
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math>
  
 
[[2014 AMC 10A  Problems/Problem 7|Solution]]
 
[[2014 AMC 10A  Problems/Problem 7|Solution]]
Line 70: Line 68:
 
==Problem 8==
 
==Problem 8==
  
Which of the following numbers is vibing?
+
Which of the following numbers is a perfect square?
  
 
<math> \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 </math>
 
<math> \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 </math>
Line 76: Line 74:
 
[[2014 AMC 10A  Problems/Problem 8|Solution]]
 
[[2014 AMC 10A  Problems/Problem 8|Solution]]
  
==Illuminati Shill==
+
==Problem 9==
  
 
The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</math> and <math>6</math>. How long is the third altitude of the triangle?
 
The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</math> and <math>6</math>. How long is the third altitude of the triangle?
Line 119: Line 117:
 
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,grey);
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);
Line 138: Line 136:
 
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);
Line 144: Line 142:
  
 
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;
  
Line 153: Line 151:
  
 
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);
Line 172: Line 170:
 
label("$I$",I,SW);
 
label("$I$",I,SW);
 
</asy>
 
</asy>
 
 
<math> \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 </math>
 
<math> \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 </math>
  
Line 181: Line 178:
 
The <math>y</math>-intercepts, <math>A</math> and <math>B</math>, of two perpendicular lines intersecting at the point <math>D(6,8)</math> have a sum of zero. What is the area of <math>\triangle DAB</math>?
 
The <math>y</math>-intercepts, <math>A</math> and <math>B</math>, of two perpendicular lines intersecting at the point <math>D(6,8)</math> have a sum of zero. What is the area of <math>\triangle DAB</math>?
  
<math> \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72  
+
<math> \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 </math>
\qquad\textbf{(D)}\qquad\textbf{(A)}\qquad\textbf{(B)}\qquad\textbf{(O)}\qquad\textbf{(N)}\qquad\textbf{(T)}\qquad\textbf{(H)}\qquad\textbf{(E)}\qquad\textbf{(H)}\qquad\textbf{(A)}\qquad\textbf{(T)}\qquad\textbf{(E)}\qquad\textbf{(R)}\qquad\textbf{(S)}</math>
 
  
 
[[2014 AMC 10A  Problems/Problem 14|Solution]]
 
[[2014 AMC 10A  Problems/Problem 14|Solution]]
Line 188: Line 184:
 
==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 she 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
Line 210: Line 202:
 
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>
Line 249: Line 241:
  
 
==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?‮
Line 262: Line 252:
 
A square in the coordinate plane has vertices whose <math>y</math>-coordinates are <math>0</math>, <math>1</math>, <math>4</math>, and <math>5</math>. What is the area of the square?
 
A square in the coordinate plane has vertices whose <math>y</math>-coordinates are <math>0</math>, <math>1</math>, <math>4</math>, and <math>5</math>. What is the area of the square?
  
<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 \qquad\textbf{(F)}\ \int_{\prod_{i=a}^{b} f(i)}^{\int_{a}^{\int_{a}^{\oiint_V f(s,t) \,ds\,dt} x^2 dx} \prod_{i=a}^{\prod_{i=a}^{b} f(i)} f(i)} \lim_{x\to\infty} f(x)</math>
+
<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 </math>
<cmath>\int_{a}^{b} x^2 dx</cmath>
+
 
 
[[2014 AMC 10A  Problems/Problem 18|Solution]]
 
[[2014 AMC 10A  Problems/Problem 18|Solution]]
  
Line 291: Line 281:
 
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));
 
 
</asy>
 
</asy>
 
[[2014 AMC 10A  Problems/Problem 19|Solution]]
 
[[2014 AMC 10A  Problems/Problem 19|Solution]]
Line 323: Line 312:
 
<asy>
 
<asy>
 
import graph;
 
import graph;
size(10cm);
+
size(6cm);
  
 
real L = 0.05;
 
real L = 0.05;

Revision as of 17:52, 24 January 2020

Problem 1

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

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

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 Roy's cat finish eating all the cat food in the box?

$\textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}\qquad\textbf{(F)}\ \text{Option 6}$

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 $0$ points, $35\%$ scored $10$ points, $30\%$ scored $15$ points, and the rest scored $30$ 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,grey); 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, $A$ and $B$, of two perpendicular lines intersecting at the point $D(6,8)$ have a sum of zero. What is the area of $\triangle DAB$?

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

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 she 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)); [/asy] Solution

Problem 20

The product $(8)(888\dots8)$, where the second factor has $991$ digits, is an integer whose digits have a sum of $1000$. What is $991$?

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

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 $500,000$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 (ProblemsAnswer KeyResources)
Preceded by
2013 AMC 10B Problems
Followed by
2014 AMC 10B 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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS