Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
GET READY FOR THE AMC 10 WITH AoPS
Learn with outstanding instructors and top-scoring students from around the world in our AMC 10 Problem Series online course.
CHECK SCHEDULE

Difference between revisions of "2017 AMC 10B Problems"

m (→‎Problem 12)
m (→‎Problem 20)
Line 112: Line 112:
  
 
==Problem 20==
 
==Problem 20==
−
Placeholder
+
The number <math>21!=51,090,942,171,709,440,000</math> has over <math>60,000</math> positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
 +
 
 +
<math>\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}</math>
 +
 
  
 
[[2017 AMC 10B Problems/Problem 20|Solution]]
 
[[2017 AMC 10B Problems/Problem 20|Solution]]

Revision as of 12:25, 16 February 2017

2017 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

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 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 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

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?

$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Solution

Problem 2

Sofia ran $5$ laps around the $400$-meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps?

$\textbf{(A)}\ 5$ minutes and $35$ seconds $\qquad\textbf{(B)}\ 6$ minutes and $40$ seconds $\qquad\textbf{(C)}\ 7$ minutes and $5$ seconds $\qquad\textbf{(D)}\ 7$ minutes and $25$ seconds $\qquad\textbf{(E)}\ 8$ minutes and $10$ seconds

Solution

Problem 3

Real numbers $x$, $y$, and $z$ satify the inequalities $0<x<1$, $-1<y<0$, and $1<z<2$. Which of the following numbers is necessarily positive?

$\textbf{(A)}\ y+x^2\qquad\textbf{(B)}\ y+xz\qquad\textbf{(C)}\ y+y^2\qquad\textbf{(D)}\ y+2y^2\qquad\textbf{(E)}\ y+z$

Solution

Problem 4

Placeholder

Solution

Problem 5

Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50$

Solution

Problem 6

What is the largest number of solid $2\text{ in}$ by $2\text{ in}$ by $1\text{ in}$ blocks that can fit in a $3\text{ in}$ by $2\text{ in}$ by $3\text{ in}$ box?

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

Solution

Problem 7

Placeholder

Solution

Problem 8

Placeholder

Solution

Problem 9

Placeholder

Solution

Problem 10

Placeholder

Solution

Problem 11

Placeholder

Solution

Problem 12

Elmer's new car gives $50\%$ percent better fuel efficiency. However, the new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?


$\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%$

Solution

Problem 13

Placeholder

Solution

Problem 14

Placeholder

Solution

Problem 15

Placeholder

Solution

Problem 16

Placeholder

Solution

Problem 17

Placeholder

Solution

Problem 18

Placeholder

Solution

Problem 19

Placeholder

Solution

Problem 20

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

$\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$


Solution

Problem 21

Placeholder

Solution

Problem 22

The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?

$\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}$

Solution

Problem 23

Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44$

Solution

Problem 24

The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 108\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 169$

Solution

Problem 25

Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$. What was her score on the sixth test?

$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\textbf{(C)}\ 96\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 100$

Solution

See also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
2017 AMC 10A Problems 		Followed by
2018 AMC 10A 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems&oldid=83715"