**online AMC 10 Problem Series course**.

# Difference between revisions of "2019 AMC 10B Problems"

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− | The | + | ==Problem 1== |

+ | |||

+ | Alicia had two containers. The first was <math>\tfrac{5}{6}</math> full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was <math>\tfrac{3}{4}</math> full of water. What is the ratio of the volume of the first container to the volume of the second container? | ||

+ | |||

+ | <math>\textbf{(A) } \frac{5}{8} \qquad \textbf{(B) } \frac{4}{5} \qquad \textbf{(C) } \frac{7}{8} \qquad \textbf{(D) } \frac{9}{10} \qquad \textbf{(E) } \frac{11}{12}</math> | ||

+ | |||

+ | ==Problem 2== | ||

+ | |||

+ | Consider the statement, "If <math>n</math> is not prime, then <math>n-2</math> is prime." Which of the following values of <math>n</math> is a counterexample to this statement. | ||

+ | <math>\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27</math> | ||

+ | |||

+ | ==Problem 3== | ||

+ | |||

+ | In a high school with <math>500</math> students, <math>40\%</math> of the seniors play a musical instrument, while <math>30\%</math> of the non-seniors do not play a musical instrument. In all, <math>46.8\%</math> of the students do not play a musical instrument. How many non-seniors play a musical instrument? | ||

+ | <math>\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266</math> | ||

+ | |||

+ | ==Problem 4== | ||

+ | |||

+ | ==Problem 5== | ||

+ | |||

+ | ==Problem 6== | ||

+ | |||

+ | There is a real <math>n</math> such that <math>(n+1)! + (n+2)! = n! \cdot 440</math>. What is the sum of the digits of <math>n</math>? | ||

+ | |||

+ | <math>\textbf{(A) }3\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12</math> | ||

+ | |||

+ | ==Problem 7== | ||

+ | |||

+ | Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or <math>n</math> pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of <math>n</math>? | ||

+ | |||

+ | <math>\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28</math> | ||

+ | |||

+ | ==Problem 8== | ||

+ | |||

+ | ==Problem 9== | ||

+ | |||

+ | ==Problem 10== | ||

+ | |||

+ | In a given plane, points <math>A</math> and <math>B</math> are <math>10</math> units apart. How many points <math>C</math> are there in the plane such that the perimeter of <math>\triangle ABC</math> is <math>50</math> units and the area of <math>\triangle ABC</math> is <math>100</math> square units? | ||

+ | |||

+ | <math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}</math> | ||

+ | |||

+ | ==Problem 11== | ||

+ | |||

+ | ==Problem 12== | ||

+ | |||

+ | ==Problem 13== | ||

+ | |||

+ | What is the sum of all real numbers <math>x</math> for which the median of the numbers <math>4,6,8,17,</math> and <math>x</math> is equal to the mean of those five numbers? | ||

+ | |||

+ | <math>\textbf{(A) } -5 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } \frac{15}{4} \qquad\textbf{(E) } \frac{35}{4}</math> | ||

+ | |||

+ | ==Problem 14== | ||

+ | |||

+ | ==Problem 15== | ||

+ | |||

+ | ==Problem 16== | ||

+ | |||

+ | ==Problem 17== | ||

+ | |||

+ | ==Problem 18== | ||

+ | |||

+ | Henry decides one morning to do a workout, and he walks <math>\tfrac{3}{4}</math> of the way from his home to his gym. The gym is <math>2</math> kilometers away from Henry's home. At that point, he changes his mind and walks <math>\tfrac{3}{4}</math> of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks <math>\tfrac{3}{4}</math> of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked <math>\tfrac{3}{4}</math> of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point <math>A</math> kilometers from home and a point <math>B</math> kilometers from home. What is <math>|A-B|</math>? | ||

+ | |||

+ | <math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}</math> | ||

+ | |||

+ | ==Problem 19== | ||

+ | |||

+ | Let <math>S</math> be the set of all positive integer divisors of <math>100,000.</math> How many numbers are the product of two distinct elements of <math>S?</math> | ||

+ | |||

+ | <math>\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121</math> | ||

+ | |||

+ | ==Problem 20== | ||

+ | |||

+ | ==Problem 21== | ||

+ | |||

+ | Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head? | ||

+ | |||

+ | <math>\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}</math> | ||

+ | |||

+ | ==Problem 22== | ||

+ | |||

+ | ==Problem 23== | ||

+ | |||

+ | Points <math>A(6,13)</math> and <math>B(12,11)</math> lie on circle <math>\omega</math> in the plane. Suppose that the tangent lines to <math>\omega</math> at <math>A</math> and <math>B</math> intersect at a point on the <math>x</math>-axis. What is the area of <math>\omega</math>? | ||

+ | |||

+ | <math>\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) } | ||

+ | \frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}</math> | ||

+ | |||

+ | ==Problem 24== | ||

+ | |||

+ | Define a sequence recursively by <math>x_0=5</math> and | ||

+ | <cmath>x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}</cmath>for all nonnegative integers <math>n.</math> Let <math>m</math> be the least positive integer such that | ||

+ | <cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>In which of the following intervals does <math>m</math> lie? | ||

+ | |||

+ | <math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]</math> | ||

+ | |||

+ | ==Problem 25== | ||

+ | |||

+ | How many sequences of <math>0</math>s and <math>1</math>s of length <math>19</math> are there that begin with a <math>0</math>, end with a <math>0</math>, contain no two consecutive <math>0</math>s, and contain no three consecutive <math>1</math>s? | ||

+ | |||

+ | <math>\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75</math> |

## Revision as of 12:05, 14 February 2019

## 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

## Problem 1

Alicia had two containers. The first was full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was full of water. What is the ratio of the volume of the first container to the volume of the second container?

## Problem 2

Consider the statement, "If is not prime, then is prime." Which of the following values of is a counterexample to this statement.

## Problem 3

In a high school with students, of the seniors play a musical instrument, while of the non-seniors do not play a musical instrument. In all, of the students do not play a musical instrument. How many non-seniors play a musical instrument?

## Problem 4

## Problem 5

## Problem 6

There is a real such that . What is the sum of the digits of ?

## Problem 7

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of ?

## Problem 8

## Problem 9

## Problem 10

In a given plane, points and are units apart. How many points are there in the plane such that the perimeter of is units and the area of is square units?

## Problem 11

## Problem 12

## Problem 13

What is the sum of all real numbers for which the median of the numbers and is equal to the mean of those five numbers?

## Problem 14

## Problem 15

## Problem 16

## Problem 17

## Problem 18

Henry decides one morning to do a workout, and he walks of the way from his home to his gym. The gym is kilometers away from Henry's home. At that point, he changes his mind and walks of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point kilometers from home and a point kilometers from home. What is ?

## Problem 19

Let be the set of all positive integer divisors of How many numbers are the product of two distinct elements of

## Problem 20

## Problem 21

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

## Problem 22

## Problem 23

Points and lie on circle in the plane. Suppose that the tangent lines to at and intersect at a point on the -axis. What is the area of ?

## Problem 24

Define a sequence recursively by and for all nonnegative integers Let be the least positive integer such that In which of the following intervals does lie?

## Problem 25

How many sequences of s and s of length are there that begin with a , end with a , contain no two consecutive s, and contain no three consecutive s?