Difference between revisions of "2018 AMC 10B Problems"
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− | ==Problem 1== | + | == Problem 1 == |
Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? | Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? | ||
− | <math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360</math> | + | <math> |
+ | \textbf{(A) } 90 \qquad | ||
+ | \textbf{(B) } 100 \qquad | ||
+ | \textbf{(C) } 180 \qquad | ||
+ | \textbf{(D) } 200 \qquad | ||
+ | \textbf{(E) } 360 | ||
+ | </math> | ||
− | ==Problem 2== | + | == Problem 2 == |
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? | Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? | ||
− | <math>\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68</math> | + | <math> |
+ | \textbf{(A) } 64 \qquad | ||
+ | \textbf{(B) } 65 \qquad | ||
+ | \textbf{(C) } 66 \qquad | ||
+ | \textbf{(D) } 67 \qquad | ||
+ | \textbf{(E) } 68 | ||
+ | </math> | ||
− | == Problem 3== | + | == Problem 3 == |
In the expression <math>\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)</math> each blank is to be filled in with one of the digits <math>1,2,3,</math> or <math>4,</math> with each digit being used once. How many different values can be obtained? | In the expression <math>\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)</math> each blank is to be filled in with one of the digits <math>1,2,3,</math> or <math>4,</math> with each digit being used once. How many different values can be obtained? | ||
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<math> | <math> | ||
\textbf{(A) }2 \qquad | \textbf{(A) }2 \qquad | ||
− | \textbf{(B) }3\qquad | + | \textbf{(B) }3 \qquad |
\textbf{(C) }4 \qquad | \textbf{(C) }4 \qquad | ||
\textbf{(D) }6 \qquad | \textbf{(D) }6 \qquad | ||
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</math> | </math> | ||
− | == Problem 4== | + | == Problem 4 == |
− | A three-dimensional rectangular box with dimensions <math>X</math>, <math>Y</math>, and <math>Z</math> has faces whose surface areas are | + | A three-dimensional rectangular box with dimensions <math>X</math>, <math>Y</math>, and <math>Z</math> has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is <math>X+Y+Z</math>? |
<math> | <math> | ||
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</math> | </math> | ||
− | ==Problem 5== | + | == Problem 5 == |
How many subsets of <math>\{2,3,4,5,6,7,8,9\}</math> contain at least one prime number? | How many subsets of <math>\{2,3,4,5,6,7,8,9\}</math> contain at least one prime number? | ||
− | <math>\textbf{(A)} | + | <math> |
+ | \textbf{(A) }128 \qquad | ||
+ | \textbf{(B) }192 \qquad | ||
+ | \textbf{(C) }224 \qquad | ||
+ | \textbf{(D) }240 \qquad | ||
+ | \textbf{(E) }256 \qquad | ||
+ | </math> | ||
+ | |||
+ | == Problem 6 == | ||
+ | |||
+ | A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required? | ||
− | + | <math> | |
+ | \textbf{(A) }\frac{1}{15} \qquad | ||
+ | \textbf{(B) }\frac{1}{10} \qquad | ||
+ | \textbf{(C) }\frac{1}{6} \qquad | ||
+ | \textbf{(D) }\frac{1}{5} \qquad | ||
+ | \textbf{(E) }\frac{1}{4} \qquad | ||
+ | </math> | ||
− | + | == Problem 7 == | |
− | <math> | + | In the figure below, <math>N</math> congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let <math>A</math> be the combined area of the small semicircles and <math>B</math> be the area of the region inside the large semicircle but outside the small semicircles. The ratio <math>A:B</math> is 1:18. What is <math>N</math> ? |
− | + | ||
+ | <math>[asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy]</math> | ||
− | + | <math> | |
+ | \textbf{(A) }16 \qquad | ||
+ | \textbf{(B) }17 \qquad | ||
+ | \textbf{(C) }18 \qquad | ||
+ | \textbf{(D) }19 \qquad | ||
+ | \textbf{(E) }36 \qquad | ||
+ | </math> | ||
+ | == Problem 8 == | ||
+ | Sara makes a staircase out of toothpicks as shown: <br> | ||
+ | <Insert diagram><br> | ||
+ | This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks? | ||
− | <math> | + | <math> |
+ | \textbf{(A) }10 \qquad | ||
+ | \textbf{(B) }11 \qquad | ||
+ | \textbf{(C) }12 \qquad | ||
+ | \textbf{(D) }24 \qquad | ||
+ | \textbf{(E) }30 \qquad | ||
+ | </math> | ||
+ | |||
+ | == Problem 9 == | ||
+ | |||
+ | The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let <math>p</math> be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability <math>p</math> ? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }13 \qquad | ||
+ | \textbf{(B) }26 \qquad | ||
+ | \textbf{(C) }32 \qquad | ||
+ | \textbf{(D) }39 \qquad | ||
+ | \textbf{(E) }42 \qquad | ||
+ | </math> | ||
+ | |||
+ | == Problem 10 == | ||
+ | |||
+ | In the rectangular parallelepiped shown, <math>AB=3</math>, <math>BC=1</math>, and <math>CG=2</math>. Point <math>M</math> is the midpoint of <math>\overline{FG}</math>. What is the volume of the rectangular pyramid with base <math>BCHE</math> and apex <math>M</math> ? <br> | ||
+ | <Insert Diagram> | ||
+ | <br> | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }1 \qquad | ||
+ | \textbf{(B) }\frac{4}{3} \qquad | ||
+ | \textbf{(C) }\frac{3}{2} \qquad | ||
+ | \textbf{(D) }\frac{5}{3} \qquad | ||
+ | \textbf{(E) }2 \qquad | ||
+ | </math> | ||
+ | |||
+ | == Problem 11 == | ||
+ | |||
+ | Which of the following expressions is never a prime number when <math>p</math> is a prime number? | ||
− | <math>\textbf{(A) } 16 \qquad \textbf{(B) } | + | <math> |
+ | \textbf{(A) }p^2+16 \qquad | ||
+ | \textbf{(B) }p^2+24 \qquad | ||
+ | \textbf{(C) }p^2+26 \qquad | ||
+ | \textbf{(D) }p^2+46 \qquad | ||
+ | \textbf{(E) }p^2+96 \qquad | ||
+ | </math> |
Revision as of 15:29, 16 February 2018
Contents
[hide]Problem 1
Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
Problem 3
In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?
Problem 4
A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is ?
Problem 5
How many subsets of contain at least one prime number?
Problem 6
A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?
Problem 7
In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is 1:18. What is ?
$[asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy]$ (Error compiling LaTeX. Unknown error_msg)
Problem 8
Sara makes a staircase out of toothpicks as shown:
<Insert diagram>
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Problem 9
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability ?
Problem 10
In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?
<Insert Diagram>
Problem 11
Which of the following expressions is never a prime number when is a prime number?