**online AMC 10 Problem Series course**.

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

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All lines with equation <math>ax+by=c</math> such that <math>a,b,c</math> form an arithmetic progression pass through a common point. What are the coordinates of that point? | All lines with equation <math>ax+by=c</math> such that <math>a,b,c</math> form an arithmetic progression pass through a common point. What are the coordinates of that point? | ||

+ | |||

+ | <math>\textbf{(A) } (-1,2) | ||

+ | \qquad\textbf{(B) } (0,1) | ||

+ | \qquad\textbf{(C) } (1,-2) | ||

+ | \qquad\textbf{(D) } (1,0) | ||

+ | \qquad\textbf{(E) } (1,2)</math> | ||

[[2019 AMC 10B Problems/Problem 4|Solution]] | [[2019 AMC 10B Problems/Problem 4|Solution]] | ||

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==Problem 9== | ==Problem 9== | ||

+ | |||

+ | The function <math>f</math> is defined by <cmath>f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|</cmath>for all real numbers <math>x</math>, where <math>\lfloor r \rfloor</math> denotes the greatest integer less than or equal to the real number <math>r</math>. What is the range of <math>f</math>? | ||

+ | |||

+ | <math>\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\} \qquad\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} </math> | ||

[[2019 AMC 10B Problems/Problem 9|Solution]] | [[2019 AMC 10B Problems/Problem 9|Solution]] | ||

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==Problem 12== | ==Problem 12== | ||

+ | |||

+ | What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than <math>2019</math>? | ||

+ | |||

+ | <math>\textbf{(A) } 11 | ||

+ | \qquad\textbf{(B) } 14 | ||

+ | \qquad\textbf{(C) } 22 | ||

+ | \qquad\textbf{(D) } 23 | ||

+ | \qquad\textbf{(E) } 27</math> | ||

[[2019 AMC 10B Problems/Problem 12|Solution]] | [[2019 AMC 10B Problems/Problem 12|Solution]] | ||

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==Problem 16== | ==Problem 16== | ||

+ | |||

+ | In <math>\triangle ABC</math> with a right angle at <math>C,</math> point <math>D</math> lies in the interior of <math>\overline{AB}</math> and point <math>E</math> lies in the interior of <math>\overline{BC}</math> so that <math>AC=CD,</math> <math>DE=EB,</math> and the ratio <math>AC:DE=4:3.</math> What is the ratio <math>AD:DB?</math> | ||

+ | |||

+ | <math>\textbf{(A) } 2:3 | ||

+ | \qquad\textbf{(B) } 2:\sqrt{5} | ||

+ | \qquad\textbf{(C) } 1:1 | ||

+ | \qquad\textbf{(D) } 3:\sqrt{5} | ||

+ | \qquad\textbf{(E) } 3:2</math> | ||

[[2019 AMC 10B Problems/Problem 16|Solution]] | [[2019 AMC 10B Problems/Problem 16|Solution]] | ||

==Problem 17== | ==Problem 17== | ||

+ | |||

+ | A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin <math>k</math> is <math>2^{-k}</math> for <math>k=1,2,3,\ldots.</math> What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | ||

+ | |||

+ | <math>\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}</math> | ||

[[2019 AMC 10B Problems/Problem 17|Solution]] | [[2019 AMC 10B Problems/Problem 17|Solution]] | ||

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==Problem 20== | ==Problem 20== | ||

+ | |||

+ | As shown in the figure, line segment <math>\overline{AD}</math> is trisected by points <math>B</math> and <math>C</math> so that <math>AB=BC=CD=2.</math> Three semicircles of radius <math>1,</math> <math>\overarc{AEB},\overarc{BFC},</math> and <math>\overarc{CGD},</math> have their diameters on <math>\overline{AD},</math> and are tangent to line <math>EG</math> at <math>E,F,</math> and <math>G,</math> respectively. A circle of radius <math>2</math> has its center on <math>F. </math> The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form | ||

+ | <cmath>\frac{a}{b}\cdot\pi-\sqrt{c}+d,</cmath>where <math>a,b,c,</math> and <math>d</math> are positive integers and <math>a</math> and <math>b</math> are relatively prime. What is <math>a+b+c+d</math>? | ||

+ | |||

+ | [asy] | ||

+ | size(6cm); | ||

+ | filldraw(circle((0,0),2), gray(0.7)); | ||

+ | filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); | ||

+ | filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); | ||

+ | filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); | ||

+ | dot((-3,-1)); | ||

+ | label("<math>A</math>",(-3,-1),S); | ||

+ | dot((-2,0)); | ||

+ | label("<math>E</math>",(-2,0),NW); | ||

+ | dot((-1,-1)); | ||

+ | label("<math>B</math>",(-1,-1),S); | ||

+ | dot((0,0)); | ||

+ | label("<math>F</math>",(0,0),N); | ||

+ | dot((1,-1)); | ||

+ | label("<math>C</math>",(1,-1), S); | ||

+ | dot((2,0)); | ||

+ | label("<math>G</math>", (2,0),NE); | ||

+ | dot((3,-1)); | ||

+ | label("<math>D</math>", (3,-1), S); | ||

+ | [/asy] | ||

+ | <math>\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17</math> | ||

[[2019 AMC 10B Problems/Problem 20|Solution]] | [[2019 AMC 10B Problems/Problem 20|Solution]] |

## Revision as of 13:23, 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

All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?

## 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

The function is defined by for all real numbers , where denotes the greatest integer less than or equal to the real number . What is the range of ?

## 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

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than ?

## 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

In with a right angle at point lies in the interior of and point lies in the interior of so that and the ratio What is the ratio

## Problem 17

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin is for What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

## 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

As shown in the figure, line segment is trisected by points and so that Three semicircles of radius and have their diameters on and are tangent to line at and respectively. A circle of radius has its center on The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form where and are positive integers and and are relatively prime. What is ?

[asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("",(-3,-1),S); dot((-2,0)); label("",(-2,0),NW); dot((-1,-1)); label("",(-1,-1),S); dot((0,0)); label("",(0,0),N); dot((1,-1)); label("",(1,-1), S); dot((2,0)); label("", (2,0),NE); dot((3,-1)); label("", (3,-1), S); [/asy]

## 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?