Difference between revisions of "2011 AMC 10A Problems"
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+ | {{AMC10 Problems|year=2011|ab=A}} | ||
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== Problem 1 == | == Problem 1 == | ||
− | A cell phone plan costs <math>\ | + | A cell phone plan costs <math>\textdollar 20</math> each month, plus <math>5</math>¢ per text message sent, plus <math>10</math>¢ for each minute used over <math>30</math> hours. In January Juan sent <math>100</math> text messages and talked for <math>30.5</math> hours. How much did he have to pay? |
− | <math> \textbf{(A)}\ | + | <math> \textbf{(A)}\ \textdollar 24.00 \qquad\textbf{(B)}\ \textdollar 24.50 \qquad\textbf{(C)}\ \textdollar 25.50\qquad\textbf{(D)}\ \textdollar 28.00\qquad\textbf{(E)}\ \textdollar 30.00 </math> |
[[2011 AMC 10A Problems/Problem 1|Solution]] | [[2011 AMC 10A Problems/Problem 1|Solution]] | ||
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== Problem 3 == | == Problem 3 == | ||
− | Suppose | + | Suppose <math>[a\ b]</math> denotes the average of <math>a</math> and <math>b</math>, and <math>\{a\ b\ c\}</math> denotes the average of <math>a, b</math>, and <math>c</math>. What is <math>\{\{1\ 1\ 0\}\ [0\ 1]\ 0\}</math>? |
<math> \textbf{(A)}\ \frac{2}{9} \qquad\textbf{(B)}\ \frac{5}{18} \qquad\textbf{(C)}\ \frac{1}{3} \qquad\textbf{(D)}\ \frac{7}{18} \qquad\textbf{(E)}\ \frac{2}{3} </math> | <math> \textbf{(A)}\ \frac{2}{9} \qquad\textbf{(B)}\ \frac{5}{18} \qquad\textbf{(C)}\ \frac{1}{3} \qquad\textbf{(D)}\ \frac{7}{18} \qquad\textbf{(E)}\ \frac{2}{3} </math> | ||
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Which of the following equations does NOT have a solution? | Which of the following equations does NOT have a solution? | ||
− | <math>\ | + | <math>\textbf{(A)}\:(x+7)^2=0</math> |
− | <math>\ | + | <math>\textbf{(B)}\:\left|-3x\right|+5=0</math> |
− | <math>\ | + | <math>\textbf{(C)}\:\sqrt{-x}-2=0</math> |
− | <math>\ | + | <math>\textbf{(D)}\:\sqrt{x}-8=0</math> |
− | <math>\ | + | <math>\textbf{(E)}\:\left|-3x\right|-4=0 </math> |
[[2011 AMC 10A Problems/Problem 7|Solution]] | [[2011 AMC 10A Problems/Problem 7|Solution]] | ||
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== Problem 8 == | == Problem 8 == | ||
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? | Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? | ||
− | |||
<math> \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60 </math> | <math> \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60 </math> | ||
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A rectangular region is bounded by the graphs of the equations <math>y=a, y=-b, x=-c,</math> and <math>x=d</math>, where <math>a,b,c,</math> and <math>d</math> are all positive numbers. Which of the following represents the area of this region? | A rectangular region is bounded by the graphs of the equations <math>y=a, y=-b, x=-c,</math> and <math>x=d</math>, where <math>a,b,c,</math> and <math>d</math> are all positive numbers. Which of the following represents the area of this region? | ||
− | <math> \textbf{(A)}\ ac+ad+bc+bd\qquad\textbf{(B)}\ ac-ad+bc-bd\qquad\textbf{(C)}\ ac+ad-bc-bd \quad\quad\qquad\textbf{(D)}\ -ac-ad+bc+bd\qquad\textbf{(E)}\ ac-ad-bc+bd </math> | + | <math> \textbf{(A)}\ ac+ad+bc+bd\qquad\textbf{(B)}\ ac-ad</math> <math>+bc-bd\qquad\textbf{(C)}\ ac+ad</math> <math>-bc-bd \quad\quad\qquad\textbf{(D)}\ -ac-ad</math> <math>+bc+bd\qquad\textbf{(E)}\ ac-ad</math> <math> -bc+bd </math> |
[[2011 AMC 10A Problems/Problem 9|Solution]] | [[2011 AMC 10A Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | A majority of the 30 students in Ms. Deameanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was <math>\ | + | A majority of the 30 students in Ms. Deameanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was <math> \textdollar 17.71</math>. What was the cost of a pencil in cents? |
− | <math>\ | + | <math>\textbf{(A)}\,7 \qquad\textbf{(B)}\,11 \qquad\textbf{(C)}\,17 \qquad\textbf{(D)}\,23 \qquad\textbf{(E)}\,77</math> |
[[2011 AMC 10A Problems/Problem 10|Solution]] | [[2011 AMC 10A Problems/Problem 10|Solution]] | ||
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Square <math>EFGH</math> has one vertex on each side of square <math>ABCD</math>. Point <math>E</math> is on <math>\overline{AB}</math> with <math>AE=7\cdot EB</math>. What is the ratio of the area of <math>EFGH</math> to the area of <math>ABCD</math>? | Square <math>EFGH</math> has one vertex on each side of square <math>ABCD</math>. Point <math>E</math> is on <math>\overline{AB}</math> with <math>AE=7\cdot EB</math>. What is the ratio of the area of <math>EFGH</math> to the area of <math>ABCD</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} </math> |
[[2011 AMC 10A Problems/Problem 11|Solution]] | [[2011 AMC 10A Problems/Problem 11|Solution]] | ||
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The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? | The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? | ||
− | <math>\ | + | <math>\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17</math> |
− | |||
[[2011 AMC 10A Problems/Problem 12|Solution]] | [[2011 AMC 10A Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
− | How many even integers are there between 200 and 700 whose digits are all different and come from the set {1,2,5,7,8,9}? | + | How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}? |
− | <math>\ | + | <math>\textbf{(A)}\,12 \qquad\textbf{(B)}\,20 \qquad\textbf{(C)}\,72 \qquad\textbf{(D)}\,120 \qquad\textbf{(E)}\,200</math> |
[[2011 AMC 10A Problems/Problem 13|Solution]] | [[2011 AMC 10A Problems/Problem 13|Solution]] | ||
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A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | ||
− | <math>\ | + | <math>\textbf{(A)}\,\frac{1}{36} \qquad\textbf{(B)}\,\frac{1}{12} \qquad\textbf{(C)}\,\frac{1}{6} \qquad\textbf{(D)}\,\frac{1}{4} \qquad\textbf{(E)}\,\frac{5}{18}</math> |
[[2011 AMC 10A Problems/Problem 14|Solution]] | [[2011 AMC 10A Problems/Problem 14|Solution]] | ||
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Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles? | Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles? | ||
− | <math>\ | + | <math>\textbf{(A)}\,140 \qquad\textbf{(B)}\,240 \qquad\textbf{(C)}\,440 \qquad\textbf{(D)}\,640 \qquad\textbf{(E)}\,840</math> |
[[2011 AMC 10A Problems/Problem 15|Solution]] | [[2011 AMC 10A Problems/Problem 15|Solution]] | ||
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Which of the following is equal to <math>\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}</math>? | Which of the following is equal to <math>\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\,3\sqrt2 \qquad\textbf{(B)}\,2\sqrt6 \qquad\textbf{(C)}\,\frac{7\sqrt2}{2} \qquad\textbf{(D)}\,3\sqrt3 \qquad\textbf{(E)}\,6</math> |
[[2011 AMC 10A Problems/Problem 16|Solution]] | [[2011 AMC 10A Problems/Problem 16|Solution]] | ||
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In the eight-term sequence <math>A,B,C,D,E,F,G,H</math>, the value of <math>C</math> is 5 and the sum of any three consecutive terms is 30. What is <math>A+H</math>? | In the eight-term sequence <math>A,B,C,D,E,F,G,H</math>, the value of <math>C</math> is 5 and the sum of any three consecutive terms is 30. What is <math>A+H</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\,17 \qquad\textbf{(B)}\,18 \qquad\textbf{(C)}\,25 \qquad\textbf{(D)}\,26 \qquad\textbf{(E)}\,43</math> |
[[2011 AMC 10A Problems/Problem 17|Solution]] | [[2011 AMC 10A Problems/Problem 17|Solution]] | ||
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\textbf{(C)}\ 2 \qquad | \textbf{(C)}\ 2 \qquad | ||
\textbf{(D)}\ \frac{3\pi}{4} \qquad | \textbf{(D)}\ \frac{3\pi}{4} \qquad | ||
− | \textbf{(E)}\ 1+\frac{\pi}{2 | + | \textbf{(E)}\ 1+\frac{\pi}{2} </math> |
[[2011 AMC 10A Problems/Problem 18|Solution]] | [[2011 AMC 10A Problems/Problem 18|Solution]] | ||
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== Problem 20 == | == Problem 20 == | ||
− | Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? | + | Two points on the circumference of a circle of radius <math>r</math> are selected independently and at random. From each point a chord of length <math>r</math> is drawn in a clockwise direction. What is the probability that the two chords intersect? |
<math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} </math> | <math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} </math> | ||
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Seven students count from 1 to 1000 as follows: | Seven students count from 1 to 1000 as follows: | ||
− | •Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000. | + | •Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, ..., 997, 999, 1000. |
•Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. | •Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. | ||
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Let <math>R</math> be a square region and <math>n\ge4</math> an integer. A point <math>X</math> in the interior of <math>R</math> is called <math>n\text{-}ray</math> partitional if there are <math>n</math> rays emanating from <math>X</math> that divide <math>R</math> into <math>n</math> triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? | Let <math>R</math> be a square region and <math>n\ge4</math> an integer. A point <math>X</math> in the interior of <math>R</math> is called <math>n\text{-}ray</math> partitional if there are <math>n</math> rays emanating from <math>X</math> that divide <math>R</math> into <math>n</math> triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? | ||
− | <math>\ | + | <math>\textbf{(A)}\,1500 \qquad\textbf{(B)}\,1560 \qquad\textbf{(C)}\,2320 \qquad\textbf{(D)}\,2480 \qquad\textbf{(E)}\,2500</math> |
[[2011 AMC 10A Problems/Problem 25|Solution]] | [[2011 AMC 10A Problems/Problem 25|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | {{AMC10 box|year=2011|ab=A|before=[[2010 AMC 10B Problems]]|after=[[2011 AMC 10B Problems]]}} | ||
+ | * [[AMC 10]] | ||
+ | * [[AMC 10 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 18:09, 19 April 2021
2011 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
A cell phone plan costs each month, plus ¢ per text message sent, plus ¢ for each minute used over hours. In January Juan sent text messages and talked for hours. How much did he have to pay?
Problem 2
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
Problem 3
Suppose denotes the average of and , and denotes the average of , and . What is ?
Problem 4
Let and be the following sums of arithmetic sequences: What is the value of ?
Problem 5
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of , , and minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
Problem 6
Set has 20 elements, and set has 15 elements. What is the smallest possible number of elements in , the union of and ?
Problem 7
Which of the following equations does NOT have a solution?
Problem 8
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
Problem 9
A rectangular region is bounded by the graphs of the equations and , where and are all positive numbers. Which of the following represents the area of this region?
Problem 10
A majority of the 30 students in Ms. Deameanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was . What was the cost of a pencil in cents?
Problem 11
Square has one vertex on each side of square . Point is on with . What is the ratio of the area of to the area of ?
Problem 12
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
Problem 13
How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}?
Problem 14
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
Problem 15
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
Problem 16
Which of the following is equal to ?
Problem 17
In the eight-term sequence , the value of is 5 and the sum of any three consecutive terms is 30. What is ?
Problem 18
Circles and each have radius 1. Circles and share one point of tangency. Circle has a point of tangency with the midpoint of . What is the area inside Circle but outside circle and circle ?
Problem 19
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
Problem 20
Two points on the circumference of a circle of radius are selected independently and at random. From each point a chord of length is drawn in a clockwise direction. What is the probability that the two chords intersect?
Problem 21
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
Problem 22
Each vertex of convex pentagon is to be assigned a color. There are colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
Problem 23
Seven students count from 1 to 1000 as follows:
•Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, ..., 997, 999, 1000.
•Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
•Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
•Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
•Finally, George says the only number that no one else says.
What number does George say?
Problem 24
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
Problem 25
Let be a square region and an integer. A point in the interior of is called partitional if there are rays emanating from that divide into triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
See also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2010 AMC 10B Problems |
Followed by 2011 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.