Difference between revisions of "2004 AMC 10B Problems"
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== Problem 1 == | == Problem 1 == | ||
− | Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club? | + | Each row of the Misty Moon Amphitheater has <math>33</math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club? |
<math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726 </math> | <math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726 </math> | ||
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== Problem 2 == | == Problem 2 == | ||
− | How many two-digit positive integers have at least one 7 as a digit? | + | How many two-digit positive integers have at least one <math>7 </math>as a digit? |
<math> \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30 </math> | <math> \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30 </math> | ||
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== Problem 3 == | == Problem 3 == | ||
− | At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice? | + | At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made <math>48</math> free throws. How many free throws did she make at the first practice? |
<math> \mathrm{(A) \ } 3 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ } 15 </math> | <math> \mathrm{(A) \ } 3 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ } 15 </math> | ||
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== Problem 4 == | == Problem 4 == | ||
− | A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P? | + | A standard six-sided die is rolled, and <math>P</math> is the product of the five numbers that are visible. What is the largest number that is certain to divide <math>P</math>? |
<math> \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720 </math> | <math> \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720 </math> | ||
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== Problem 8 == | == Problem 8 == | ||
− | Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? | + | Minneapolis-St. Paul International Airport is <math>8</math> miles southwest of downtown St. Paul and <math>10</math> miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? |
<math> \mathrm{(A) \ } 13 \qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 17 </math> | <math> \mathrm{(A) \ } 13 \qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 17 </math> | ||
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== Problem 9 == | == Problem 9 == | ||
− | A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? | + | A square has sides of length <math>10</math>, and a circle centered at one of its vertices has radius <math>10</math>. What is the area of the union of the regions enclosed by the square and the circle? |
<math> \mathrm{(A) \ } 200+25\pi \qquad \mathrm{(B) \ } 100+75\pi \qquad \mathrm{(C) \ } 75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi </math> | <math> \mathrm{(A) \ } 200+25\pi \qquad \mathrm{(B) \ } 100+75\pi \qquad \mathrm{(C) \ } 75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi </math> | ||
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== Problem 10 == | == Problem 10 == | ||
− | A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain? | + | A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains <math>100</math> cans, how many rows does it contain? |
<math> \mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 411 </math> | <math> \mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 411 </math> | ||
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== Problem 11 == | == Problem 11 == | ||
− | Two eight-sided dice each have faces numbered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? | + | Two eight-sided dice each have faces numbered <math>1</math> through <math>8</math>. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? |
<math> \mathrm{(A) \ } \frac{1}{2} \qquad \mathrm{(B) \ } \frac{47}{64} \qquad \mathrm{(C) \ } \frac{3}{4} \qquad \mathrm{(D) \ } \frac{55}{64} \qquad \mathrm{(E) \ } \frac{7}{8} </math> | <math> \mathrm{(A) \ } \frac{1}{2} \qquad \mathrm{(B) \ } \frac{47}{64} \qquad \mathrm{(C) \ } \frac{3}{4} \qquad \mathrm{(D) \ } \frac{55}{64} \qquad \mathrm{(E) \ } \frac{7}{8} </math> | ||
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Patty has <math>20</math> coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have <math>70</math> cents more. How much are her coins worth? | Patty has <math>20</math> coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have <math>70</math> cents more. How much are her coins worth? | ||
− | <math> \mathrm{(A) \ } | + | <math> \mathrm{(A) \ } $ </math>1.15 \qquad \mathrm{(B) \ } $ <math>1.20 \qquad \mathrm{(C) \ } $ </math>1.25 \qquad \mathrm{(D) \ } $ <math>1.30 \qquad \mathrm{(E) \ } $ </math>1.35 <math> |
[[2004 AMC 10B Problems/Problem 15|Solution]] | [[2004 AMC 10B Problems/Problem 15|Solution]] | ||
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== Problem 16 == | == Problem 16 == | ||
− | Three circles of radius <math>1< | + | Three circles of radius </math>1<math> are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? |
− | <math> \mathrm{(A) \ } \frac{2 + \sqrt{6}}{3} \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } \frac{2 + 3\sqrt{2}}{2} \qquad \mathrm{(D) \ } \frac{3 + 2\sqrt{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} < | + | </math> \mathrm{(A) \ } \frac{2 + \sqrt{6}}{3} \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } \frac{2 + 3\sqrt{2}}{2} \qquad \mathrm{(D) \ } \frac{3 + 2\sqrt{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} <math> |
[[2004 AMC 10B Problems/Problem 16|Solution]] | [[2004 AMC 10B Problems/Problem 16|Solution]] | ||
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The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? | The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? | ||
− | <math> \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 < | + | </math> \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 <math> |
[[2004 AMC 10B Problems/Problem 17|Solution]] | [[2004 AMC 10B Problems/Problem 17|Solution]] | ||
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== Problem 18 == | == Problem 18 == | ||
− | In the right triangle <math>\triangle ACE< | + | In the right triangle </math>\triangle ACE<math>, we have </math>AC=12<math>, </math>CE=16<math>, and </math>EA=20<math>. Points </math>B<math>, </math>D<math>, and </math>F<math> are located on </math>AC<math>, </math>CE<math>, and </math>EA<math>, respectively, so that </math>AB=3<math>, </math>CD=4<math>, and </math>EF=5<math>. What is the ratio of the area of </math>\triangle DBF<math> to that of </math>\triangle ACE<math>? |
− | <math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{9}{25} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{11}{25} \qquad \mathrm{(E) \ } \frac{7}{16} | + | </math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{9}{25} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{11}{25} \qquad \mathrm{(E) \ } \frac{7}{16} $ |
<asy> | <asy> | ||
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== Problem 22 == | == Problem 22 == | ||
− | A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? | + | A triangle with sides of <math>5, 12,</math> and <math>13</math> has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? |
<math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{(C) \ } \sqrt{15} \qquad \mathrm{(D) \ } \frac{\sqrt{65}}{2} \qquad \mathrm{(E) \ } \frac{9}{2} </math> | <math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{(C) \ } \sqrt{15} \qquad \mathrm{(D) \ } \frac{\sqrt{65}}{2} \qquad \mathrm{(E) \ } \frac{9}{2} </math> | ||
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== Problem 23 == | == Problem 23 == | ||
− | Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? | + | Each face of a cube is painted either red or blue, each with probability <math>1/2</math>. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? |
<math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2} </math> | <math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2} </math> |
Revision as of 15:25, 4 June 2011
Contents
[hide]- 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 19
- 17 Problem 20
- 18 Problem 21
- 19 Problem 22
- 20 Problem 23
- 21 Problem 24
- 22 Problem 25
- 23 See also
Problem 1
Each row of the Misty Moon Amphitheater has seats. Rows through are reserved for a youth club. How many seats are reserved for this club?
Problem 2
How many two-digit positive integers have at least one as a digit?
Problem 3
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made free throws. How many free throws did she make at the first practice?
Problem 4
A standard six-sided die is rolled, and is the product of the five numbers that are visible. What is the largest number that is certain to divide ?
Problem 5
In the expression , the values of , , , and are , , , and , although not necessarily in that order. What is the maximum possible value of the result?
Problem 6
Which of the following numbers is a perfect square?
Problem 7
On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving Canadian dollars for every U.S. dollars. After spending Canadian dollars, she had Canadian dollars left. What is the sum of the digits of ?
Problem 8
Minneapolis-St. Paul International Airport is miles southwest of downtown St. Paul and miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
Problem 9
A square has sides of length , and a circle centered at one of its vertices has radius . What is the area of the union of the regions enclosed by the square and the circle?
Problem 10
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains cans, how many rows does it contain?
Problem 11
Two eight-sided dice each have faces numbered through . When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum?
Problem 12
An annulus is the region between two concentric circles. The concentric circles in the figure have radii and , with . Let be a radius of the larger circle, let be tangent to the smaller circle at , and let be the radius of the larger circle that contains . Let , , and . What is the area of the annulus?
Problem 13
In the United States, coins have the following thicknesses: penny, mm; nickel, mm; dime, mm; quarter, mm. If a stack of these coins is exactly mm high, how many coins are in the stack?
Problem 14
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only of the marbles in the bag are blue. Then yellow marbles are added to the bag until only of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
Problem 15
Patty has coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have cents more. How much are her coins worth?
1.15 \qquad \mathrm{(B) \ } $ 1.25 \qquad \mathrm{(D) \ } $ 1.35 $[[2004 AMC 10B Problems/Problem 15|Solution]]
== Problem 16 ==
Three circles of radius$ (Error compiling LaTeX. Unknown error_msg)1 \mathrm{(A) \ } \frac{2 + \sqrt{6}}{3} \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } \frac{2 + 3\sqrt{2}}{2} \qquad \mathrm{(D) \ } \frac{3 + 2\sqrt{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} $[[2004 AMC 10B Problems/Problem 16|Solution]]
== Problem 17 ==
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 $[[2004 AMC 10B Problems/Problem 17|Solution]]
== Problem 18 ==
In the right triangle$ (Error compiling LaTeX. Unknown error_msg)\triangle ACEAC=12CE=16EA=20BDFACCEEAAB=3CD=4EF=5\triangle DBF\triangle ACE \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{9}{25} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{11}{25} \qquad \mathrm{(E) \ } \frac{7}{16} $
Problem 19
In the sequence , , , , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is . What is the term in this sequence?
Problem 20
In points and lie on and , respectively. If and intersect at so that and , what is ?
Problem 21
Let ; ; and ; ; be two arithmetic progressions. The set is the union of the first terms of each sequence. How many distinct numbers are in ?
Problem 22
A triangle with sides of and has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
Problem 23
Each face of a cube is painted either red or blue, each with probability . The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
Problem 24
In we have , , and . Point is on the circumscribed circle of the triangle so that bisects . What is the value of ?
Problem 25
A circle of radius is internally tangent to two circles of radius at points and , where is a diameter of the smaller circle. What is the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles?