Difference between revisions of "2009 AMC 8 Problems"

m (Problem 8)
(25 intermediate revisions by 16 users not shown)
Line 1: Line 1:
 +
{{AMC8 Problems|year=2009}}
 
==Problem 1==
 
==Problem 1==
 
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?  
 
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?  
  
 
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14 </math>
 
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14 </math>
 
[[2009 AMC 8 Problems/Problem 1|Solution]]
 
  
 
==Problem 2==
 
==Problem 2==
Line 33: Line 32:
 
label("4", (-1, 20), black);
 
label("4", (-1, 20), black);
 
dot((5,5), black+linewidth(5));
 
dot((5,5), black+linewidth(5));
dot((10,10), black+linewidth(5));
+
dot((10,10),black+linewidth(5));
dot((15, 15), black+linewidth(5));
+
dot((15,15), black+linewidth(5));
 
dot((20,20), black+linewidth(5));
 
dot((20,20), black+linewidth(5));
 
label("MINUTES", (11,-5), S);
 
label("MINUTES", (11,-5), S);
 
label(rotate(90)*"MILES", (-5,11), W);</asy>
 
label(rotate(90)*"MILES", (-5,11), W);</asy>
  
<math> \textbf{(A)}5\qquad\textbf{(B)}5.5\qquad\textbf{(C)}6\qquad\textbf{(D)}6.5\qquad\textbf{(E)}7 </math>
+
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 5.5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 6.5\qquad\textbf{(E)}\ 7 </math>
  
 
[[2009 AMC 8 Problems/Problem 3|Solution]]
 
[[2009 AMC 8 Problems/Problem 3|Solution]]
Line 77: Line 76:
 
pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)};
 
pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)};
 
pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)};
 
pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)};
 +
 
int i;
 
int i;
 
for(int i=0; i<15; i=i+1) {
 
for(int i=0; i<15; i=i+1) {
Line 84: Line 84:
 
draw(shift(d[i])*p);
 
draw(shift(d[i])*p);
 
draw(shift(e[i])*p);
 
draw(shift(e[i])*p);
}</asy>
+
}
 +
</asy>
 +
 
 +
<cmath> \textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)} </cmath>
 +
 
 +
[[2009 AMC 8 Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
Line 91: Line 96:
 
<math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ 99 </math>
 
<math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ 99 </math>
  
 +
[[2009 AMC 8 Problems/Problem 5|Solution]]
 
==Problem 6==
 
==Problem 6==
 
Steve's empty swimming pool will hold <math>24,000</math> gallons of water when full. It will be filled by <math>4</math> hoses, each of which supplies <math>2.5</math> gallons of water per minute. How many hours will it take to fill Steve's pool?   
 
Steve's empty swimming pool will hold <math>24,000</math> gallons of water when full. It will be filled by <math>4</math> hoses, each of which supplies <math>2.5</math> gallons of water per minute. How many hours will it take to fill Steve's pool?   
  
 
<math> \textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 48 </math>
 
<math> \textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 48 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
Line 131: Line 139:
  
 
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4.5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 9 </math>
 
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4.5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 9 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
 
The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?  
 
The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?  
 +
 
<math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 99\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 110 </math>
 
<math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 99\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 110 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
+
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
  
 
<asy>
 
<asy>
Line 146: Line 159:
 
draw(A--C, dashed);</asy>
 
draw(A--C, dashed);</asy>
  
<math> \textbf{(A)}21\qquad\textbf{(B)}23\qquad\textbf{(C)}25\qquad\textbf{(D)}27\qquad\textbf{(E)}29 </math>
+
<math> \textbf{(A)} \ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 29 </math>
 +
 
 +
[[2009 AMC 8 Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
Line 200: Line 215:
 
fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);</asy>
 
fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);</asy>
  
<math> \textbf{(A)}\frac{1}{16}\qquad\textbf{(B)}\frac{7}{16}\qquad\textbf{(C)}\frac{1}2\qquad\textbf{(D)}\frac{9}{16}\qquad\textbf{(E)}\frac{49}{64} </math>
+
<math> \textbf{(A)}\ \frac{1}{16}\qquad\textbf{(B)}\ \frac{7}{16}\qquad\textbf{(C)}\ \frac{1}2\qquad\textbf{(D)}\ \frac{9}{16}\qquad\textbf{(E)}\ \frac{49}{64} </math>
 +
 
 +
[[2009 AMC 8 Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
Line 207: Line 224:
 
<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>
  
 +
[[2009 AMC 8 Problems/Problem 11|Solution]]
 
==Problem 12==
 
==Problem 12==
 
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?  
 
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?  
Line 231: Line 249:
 
<math> \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{7}{9}\qquad\textbf{(E)}\ \frac{5}{6} </math>
 
<math> \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{7}{9}\qquad\textbf{(E)}\ \frac{5}{6} </math>
  
 +
[[2009 AMC 8 Problems/Problem 12|Solution]]
 
==Problem 13==
 
==Problem 13==
 
A three-digit integer contains one of each of the digits <math>1</math>, <math>3</math>, and <math>5</math>. What is the probability that the integer is divisible by <math>5</math>?
 
A three-digit integer contains one of each of the digits <math>1</math>, <math>3</math>, and <math>5</math>. What is the probability that the integer is divisible by <math>5</math>?
Line 236: Line 255:
 
<math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6} </math>
 
<math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6} </math>
  
 +
[[2009 AMC 8 Problems/Problem 13|Solution]]
 
==Problem 14==
 
==Problem 14==
Austin and Temple are <math>50</math> miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging <math>60</math> miles per hour. Leaving the car with her daughter, Bonnie rod a bus back to Austin along the same route and averaged <math>40</math> miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?   
+
Austin and Temple are <math>50</math> miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging <math>60</math> miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged <math>40</math> miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?   
  
 
<math> \textbf{(A)}\ 46\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 54 </math>
 
<math> \textbf{(A)}\ 46\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 54 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
Line 245: Line 267:
  
 
<math> \textbf{(A)}\ 5\frac{1}8\qquad\textbf{(B)}\ 6\frac{1}4\qquad\textbf{(C)}\ 7\frac{1}2\qquad\textbf{(D)}\ 8\frac{3}4\qquad\textbf{(E)}\ 9\frac{7}8 </math>
 
<math> \textbf{(A)}\ 5\frac{1}8\qquad\textbf{(B)}\ 6\frac{1}4\qquad\textbf{(C)}\ 7\frac{1}2\qquad\textbf{(D)}\ 8\frac{3}4\qquad\textbf{(E)}\ 9\frac{7}8 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
Line 250: Line 274:
  
 
<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ 24 </math>
 
<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ 24 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
Line 255: Line 281:
  
 
<math> \textbf{(A)}\ 80\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 115\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 610 </math>
 
<math> \textbf{(A)}\ 80\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 115\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 610 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
Line 282: Line 310:
  
 
<math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 126 </math>
 
<math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 126 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
Line 287: Line 317:
  
 
<math> \textbf{(A)}\ 95\qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 180 </math>
 
<math> \textbf{(A)}\ 95\qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 180 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
Line 301: Line 333:
  
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math>
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
Line 306: Line 340:
  
 
<math> \textbf{(A)}\ \frac{64}{225}\qquad\textbf{(B)}\ \frac{8}{15}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{225}{64} </math>
 
<math> \textbf{(A)}\ \frac{64}{225}\qquad\textbf{(B)}\ \frac{8}{15}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{225}{64} </math>
 +
 +
[[2009 AMC 8 Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
Line 311: Line 347:
  
 
<math> \textbf{(A)}\ 512\qquad\textbf{(B)}\ 648\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 800 </math>
 
<math> \textbf{(A)}\ 512\qquad\textbf{(B)}\ 648\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 800 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought <math>400</math> jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?   
+
On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought <math>400</math> jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?   
  
 
<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math>
 
<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
Line 321: Line 361:
  
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math>
 
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
 
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is <math>1/2</math> foot from the top face. The second cut is <math>1/3</math> foot below the first cut, and the third cut is <math>1/17</math> foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
 
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is <math>1/2</math> foot from the top face. The second cut is <math>1/3</math> foot below the first cut, and the third cut is <math>1/17</math> foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
 
<asy>
 
<asy>
 +
size(120);
 +
 
import three;
 
import three;
 
real d=11/102;
 
real d=11/102;
Line 345: Line 389:
 
label("1/17", (0,1,1/6-d/4), E);</asy>
 
label("1/17", (0,1,1/6-d/4), E);</asy>
 
<asy>
 
<asy>
 +
size(120);
 +
 
import three;
 
import three;
 
real d=11/102;
 
real d=11/102;
Line 371: Line 417:
  
 
<math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math>
 
<math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math>
 +
 +
[[2009 AMC 8 Problems/Problem 25|Solution]]
 +
 +
==See Also==
 +
{{AMC8 box|year=2009|before=[[2008 AMC 8 Problems|2008 AMC 8]]|after=[[2010 AMC 8 Problems|2010 AMC 8]]}}
 +
* [[AMC 8]]
 +
* [[AMC 8 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
 +
 +
 +
{{MAA Notice}}

Revision as of 16:32, 27 September 2020

2009 AMC 8 (Answer Key)
Printable version: Wiki | AoPS ResourcesPDF

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 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. In particular, calculators and computers are not permitted.
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 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

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14$

Problem 2

On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 112$

Solution

Problem 3

The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?


[asy] import graph; /* this is a label */ Label f;  f.p=fontsize(0); xaxis(-0.9,20,Ticks(f, 5.0, 5.0));  yaxis(-0.9,20, Ticks(f, 22.0,5.0)); // real f(real x)  {  return x; }  draw(graph(f,-1,22),black+linewidth(1)); label("1", (-1,5), black);  label("2", (-1, 10), black); label("3", (-1, 15), black); label("4", (-1, 20), black); dot((5,5), black+linewidth(5)); dot((10,10),black+linewidth(5)); dot((15,15), black+linewidth(5)); dot((20,20), black+linewidth(5)); label("MINUTES", (11,-5), S); label(rotate(90)*"MILES", (-5,11), W);[/asy]

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 5.5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 6.5\qquad\textbf{(E)}\ 7$

Solution

Problem 4

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?


[asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p);[/asy]

[asy] size(350); defaultpen(linewidth(0.6)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; pair[] a={(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2), (3,3), (3,4)}; pair[] b={(5,3), (5,4), (6,2), (6,3), (6,4), (7,1), (7,2), (7,3), (7,4), (8,0), (8,1), (8,2), (9,0), (9,1), (9,2)}; pair[] c={(11,0), (11,1), (11,2), (11,3), (11,4), (12,1), (12,2), (12,3), (12,4), (13,2), (13,3), (13,4), (14,3), (14,4), (15,4)}; pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)};  int i; for(int i=0; i<15; i=i+1) { draw(shift(a[i])*p); draw(shift(b[i])*p); draw(shift(c[i])*p); draw(shift(d[i])*p); draw(shift(e[i])*p); } [/asy]

\[\textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)}\]

Solution

Problem 5

A sequence of numbers starts with $1$, $2$, and $3$. The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?

$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ 99$

Solution

Problem 6

Steve's empty swimming pool will hold $24,000$ gallons of water when full. It will be filled by $4$ hoses, each of which supplies $2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?

$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 48$

Solution

Problem 7

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?

[asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label("$A$", A, SE); label("$B$", B, SE); label("$C$", C, SE); label("$D$", D, SE); label("$3$", (1/3,3), E); label("$3$", (1/3,9), E); label("$3$", (-3,0), S); label("Main", (-3,0), N); label(rotate(45)*"Aspen", A--C, SE); label(rotate(63.43494882)*"Brown", A--D, NW);[/asy]

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4.5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 9$

Solution

Problem 8

The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?

$\textbf{(A)}\ 90\qquad\textbf{(B)}\ 99\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 110$

Solution

Problem 9

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]

$\textbf{(A)} \ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 29$

Solution

Problem 10

On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? [asy] unitsize(10); draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); draw((1,8)--(1,0)); draw((7,8)--(7,0)); draw((6,8)--(6,0)); draw((5,8)--(5,0)); draw((4,8)--(4,0)); draw((3,8)--(3,0)); draw((2,8)--(2,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black); fill((4,0)--(5,0)--(5,1)--(4,1)--cycle,black); fill((6,0)--(7,0)--(7,1)--(6,1)--cycle,black); fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); fill((4,2)--(5,2)--(5,3)--(4,3)--cycle,black); fill((6,2)--(7,2)--(7,3)--(6,3)--cycle,black); fill((0,4)--(1,4)--(1,5)--(0,5)--cycle,black); fill((2,4)--(3,4)--(3,5)--(2,5)--cycle,black); fill((4,4)--(5,4)--(5,5)--(4,5)--cycle,black); fill((6,4)--(7,4)--(7,5)--(6,5)--cycle,black); fill((0,6)--(1,6)--(1,7)--(0,7)--cycle,black); fill((2,6)--(3,6)--(3,7)--(2,7)--cycle,black); fill((4,6)--(5,6)--(5,7)--(4,7)--cycle,black); fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black); fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,black); fill((5,1)--(6,1)--(6,2)--(5,2)--cycle,black); fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,black); fill((1,3)--(2,3)--(2,4)--(1,4)--cycle,black); fill((3,3)--(4,3)--(4,4)--(3,4)--cycle,black); fill((5,3)--(6,3)--(6,4)--(5,4)--cycle,black); fill((7,3)--(8,3)--(8,4)--(7,4)--cycle,black); fill((1,5)--(2,5)--(2,6)--(1,6)--cycle,black); fill((3,5)--(4,5)--(4,6)--(3,6)--cycle,black); fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black); fill((7,5)--(8,5)--(8,6)--(7,6)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black); fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black); fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);[/asy]

$\textbf{(A)}\ \frac{1}{16}\qquad\textbf{(B)}\ \frac{7}{16}\qquad\textbf{(C)}\ \frac{1}2\qquad\textbf{(D)}\ \frac{9}{16}\qquad\textbf{(E)}\ \frac{49}{64}$

Solution

Problem 11

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$. How many more sixth graders than seventh graders bought a pencil?

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

Solution

Problem 12

The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?

[asy] unitsize(30);  draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$1$",(0,.5)); label("$3$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$5$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy] [asy] unitsize(30);  draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$2$",(0,.5)); label("$4$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$6$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy]

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{7}{9}\qquad\textbf{(E)}\ \frac{5}{6}$

Solution

Problem 13

A three-digit integer contains one of each of the digits $1$, $3$, and $5$. What is the probability that the integer is divisible by $5$?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$

Solution

Problem 14

Austin and Temple are $50$ miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging $60$ miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged $40$ miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?

$\textbf{(A)}\ 46\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 54$

Solution

Problem 15

A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, $1/4$ cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?

$\textbf{(A)}\ 5\frac{1}8\qquad\textbf{(B)}\ 6\frac{1}4\qquad\textbf{(C)}\ 7\frac{1}2\qquad\textbf{(D)}\ 8\frac{3}4\qquad\textbf{(E)}\ 9\frac{7}8$

Solution

Problem 16

How many 3-digit positive integers have digits whose product equals $24$?

$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ 24$

Solution

Problem 17

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$?

$\textbf{(A)}\ 80\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 115\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 610$

Solution

Problem 18

The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed?

[asy] unitsize(10); draw((0,0)--(7,0)--(7,7)--(0,7)--cycle); draw((1,7)--(1,0)); draw((6,7)--(6,0)); draw((5,7)--(5,0)); draw((4,7)--(4,0)); draw((3,7)--(3,0)); draw((2,7)--(2,0)); draw((0,1)--(7,1)); draw((0,2)--(7,2)); draw((0,3)--(7,3)); draw((0,4)--(7,4)); draw((0,5)--(7,5)); draw((0,6)--(7,6)); fill((1,0)--(2,0)--(2,7)--(1,7)--cycle,black); fill((3,0)--(4,0)--(4,7)--(3,7)--cycle,black); fill((5,0)--(6,0)--(6,7)--(5,7)--cycle,black); fill((0,5)--(0,6)--(7,6)--(7,5)--cycle,black); fill((0,3)--(0,4)--(7,4)--(7,3)--cycle,black); fill((0,1)--(0,2)--(7,2)--(7,1)--cycle,black);[/asy]

$\textbf{(A)}\ 49\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 126$

Solution

Problem 19

Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$?

$\textbf{(A)}\ 95\qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 180$

Solution

Problem 20

How many non-congruent triangles have vertices at three of the eight points in the array shown below? [asy] dot((0,0)); dot((0,.5)); dot((.5,0)); dot((.5,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy]

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

Problem 21

Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\frac{A}{B}$?

$\textbf{(A)}\ \frac{64}{225}\qquad\textbf{(B)}\ \frac{8}{15}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{225}{64}$

Solution

Problem 22

How many whole numbers between 1 and 1000 do not contain the digit 1?

$\textbf{(A)}\ 512\qquad\textbf{(B)}\ 648\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 800$

Solution

Problem 23

On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?

$\textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34$

Solution

Problem 24

The letters $A$, $B$, $C$ and $D$ represent digits. If $\begin{tabular}{ccc}&A&B\\ +&C&A\\ \hline &D&A\end{tabular}$ and $\begin{tabular}{ccc}&A&B\\ -&C&A\\ \hline &&A\end{tabular}$, what digit does $D$ represent?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

Problem 25

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $1/2$ foot from the top face. The second cut is $1/3$ foot below the first cut, and the third cut is $1/17$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? [asy] size(120);  import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label("A", (1,0,3/4), W); label("B", (1,0,1/3), W); label("C", (1,0,1/6-d/4), W); label("D", (1,0,d/2), W); label("1/2", (1,1,3/4), E); label("1/3", (1,1,1/3), E); label("1/17", (0,1,1/6-d/4), E);[/asy] [asy] size(120);  import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label("D", (1/2, 1, 0), SE); label("A", (1+1/2, 1, 0), SE); label("B", (2+1/2, 1, 0), SE); label("C", (3+1/2, 1, 0), SE);[/asy]

$\textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11$

Solution

See Also

2009 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
2008 AMC 8
Followed by
2010 AMC 8
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 AJHSME/AMC 8 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