Difference between revisions of "2019 AMC 10B Problems"
(→Problem 20) |
Xiazhang01 (talk | contribs) (→Problem 1) |
||
(54 intermediate revisions by 17 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AMC10 Problems|year=2019|ab=B}} | ||
+ | |||
==Problem 1== | ==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? | + | 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> | <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> | ||
Line 9: | Line 11: | ||
==Problem 2== | ==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 | + | 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> | <math>\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27</math> | ||
Line 36: | Line 38: | ||
==Problem 5== | ==Problem 5== | ||
+ | |||
+ | Triangle <math>ABC</math> lies in the first quadrant. Points <math>A</math>, <math>B</math>, and <math>C</math> are reflected across the line <math>y=x</math> to points <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Assume that none of the vertices of the triangle lie on the line <math>y=x</math>. Which of the following statements is <i><u>not</u></i> always true? | ||
+ | |||
+ | <math>\textbf{(A) }</math> Triangle <math>A'B'C'</math> lies in the first quadrant. | ||
+ | |||
+ | <math>\textbf{(B) }</math> Triangles <math>ABC</math> and <math>A'B'C'</math> have the same area. | ||
+ | |||
+ | <math>\textbf{(C) }</math> The slope of line <math>AA'</math> is <math>-1</math>. | ||
+ | |||
+ | <math>\textbf{(D) }</math> The slopes of lines <math>AA'</math> and <math>CC'</math> are the same. | ||
+ | |||
+ | <math>\textbf{(E) }</math> Lines <math>AB</math> and <math>A'B'</math> are perpendicular to each other. | ||
[[2019 AMC 10B Problems/Problem 5|Solution]] | [[2019 AMC 10B Problems/Problem 5|Solution]] | ||
Line 49: | Line 63: | ||
==Problem 7== | ==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>? | + | Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either <math>12</math> pieces of red candy, <math>14</math> pieces of green candy, <math>15</math> pieces of blue candy, or <math>n</math> pieces of purple candy. A piece of purple candy costs <math>20</math> 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> | <math>\textbf{(A) } 18 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 24\qquad \textbf{(D) } 25 \qquad \textbf{(E) } 28</math> | ||
Line 56: | Line 70: | ||
==Problem 8== | ==Problem 8== | ||
+ | |||
+ | The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length <math>2</math> and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? | ||
+ | |||
+ | <asy> | ||
+ | pen white = gray(1); | ||
+ | pen gray = gray(0.5); | ||
+ | draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle); | ||
+ | fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray); | ||
+ | draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle); | ||
+ | fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white); | ||
+ | draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle); | ||
+ | fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white); | ||
+ | draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle); | ||
+ | fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white); | ||
+ | draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle); | ||
+ | fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white); | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) } 4 \qquad \textbf{(B) } 12 - 4\sqrt{3} \qquad \textbf{(C) } 3\sqrt{3}\qquad \textbf{(D) } 4\sqrt{3} \qquad \textbf{(E) } 16 - 4\sqrt{3}</math> | ||
[[2019 AMC 10B Problems/Problem 8|Solution]] | [[2019 AMC 10B Problems/Problem 8|Solution]] | ||
Line 63: | Line 96: | ||
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>? | 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\} | + | <math>\textbf{(A) }</math> <math>\{-1, 0\}</math> |
+ | |||
+ | <math>\textbf{(B) }</math> <math>\text{The set of nonpositive integers} </math> | ||
+ | |||
+ | <math>\textbf{(C) }</math> <math>\{-1, 0, 1\}</math> | ||
+ | |||
+ | <math>\textbf{(D) }</math> <math>\{0\} </math> | ||
+ | |||
+ | <math>\textbf{(E) }</math> <math>\text{The set of nonnegative integers} </math> | ||
[[2019 AMC 10B Problems/Problem 9|Solution]] | [[2019 AMC 10B Problems/Problem 9|Solution]] | ||
Line 76: | Line 117: | ||
==Problem 11== | ==Problem 11== | ||
+ | Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar <math>1</math> the ratio of blue to green marbles is <math>9:1</math>, and the ratio of blue to green marbles in Jar <math>2</math> is <math>8:1</math>. There are <math>95</math> green marbles in all. How many more blue marbles are in Jar <math>1</math> than in Jar <math>2</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 5 | ||
+ | \qquad\textbf{(B) } 10 | ||
+ | \qquad\textbf{(C) } 25 | ||
+ | \qquad\textbf{(D) } 45 | ||
+ | \qquad\textbf{(E) } 50</math> | ||
[[2019 AMC 10B Problems/Problem 11|Solution]] | [[2019 AMC 10B Problems/Problem 11|Solution]] | ||
Line 100: | Line 148: | ||
==Problem 14== | ==Problem 14== | ||
+ | The base-ten representation for <math>19!</math> is <math>121,6T5,100,40M,832,H00</math>, where <math>T</math>, <math>M</math>, and <math>H</math> denote digits that are not given. What is <math>T+M+H</math>? | ||
+ | |||
+ | <math>\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17 </math> | ||
[[2019 AMC 10B Problems/Problem 14|Solution]] | [[2019 AMC 10B Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | Right triangles <math>T_1</math> and <math>T_2</math> have areas 1 and 2, respectively. A side of <math>T_1</math> is congruent to a side of <math>T_2</math>, and a different side of <math>T_1</math> is congruent to a different side of <math>T_2</math>. What is the square of the product of the other (third) sides of <math>T_1</math> and <math>T_2</math>? | ||
+ | |||
+ | <math>\textbf{(A) } \frac{28}{3} \qquad\textbf{(B) }10\qquad\textbf{(C) } \frac{32}{3} \qquad\textbf{(D) } \frac{34}{3} \qquad\textbf{(E) }12</math> | ||
[[2019 AMC 10B Problems/Problem 15|Solution]] | [[2019 AMC 10B Problems/Problem 15|Solution]] | ||
Line 182: | Line 236: | ||
==Problem 22== | ==Problem 22== | ||
+ | |||
+ | Raashan, Sylvia, and Ted play the following game. Each starts with <math> \$1</math>. A bell rings every <math>15</math> seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives <math>\$1</math> to that player. What is the probability that after the bell has rung <math>2019</math> times, each player will have <math>\$1</math>? (For example, Raashan and Ted may each decide to give <math>\$1</math> to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have <math>\$0</math>, Sylvia will have <math>\$2</math>, and Ted will have <math>\$1</math>, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their <math> \$1</math> to, and the holdings will be the same at the end of the second round.) | ||
+ | |||
+ | <math>\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}</math> | ||
[[2019 AMC 10B Problems/Problem 22|Solution]] | [[2019 AMC 10B Problems/Problem 22|Solution]] | ||
Line 200: | Line 258: | ||
<cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>In which of the following intervals does <math>m</math> lie? | <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>\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> |
[[2019 AMC 10B Problems/Problem 24|Solution]] | [[2019 AMC 10B Problems/Problem 24|Solution]] | ||
Line 211: | Line 269: | ||
[[2019 AMC 10B Problems/Problem 25|Solution]] | [[2019 AMC 10B Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | {{AMC10 box|year=2019|ab=B|before=[[2019 AMC 10A Problems]]|after=[[2020 AMC 10A Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 12:54, 1 February 2021
2019 AMC 10B (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
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 |
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
- 26 See also
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
Triangle lies in the first quadrant. Points , , and are reflected across the line to points , , and , respectively. Assume that none of the vertices of the triangle lie on the line . Which of the following statements is not always true?
Triangle lies in the first quadrant.
Triangles and have the same area.
The slope of line is .
The slopes of lines and are the same.
Lines and are perpendicular to each other.
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 pieces of red candy, pieces of green candy, pieces of blue candy, or pieces of purple candy. A piece of purple candy costs cents. What is the smallest possible value of ?
Problem 8
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?
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
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar the ratio of blue to green marbles is , and the ratio of blue to green marbles in Jar is . There are green marbles in all. How many more blue marbles are in Jar than in Jar ?
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
The base-ten representation for is , where , , and denote digits that are not given. What is ?
Problem 15
Right triangles and have areas 1 and 2, respectively. A side of is congruent to a side of , and a different side of is congruent to a different side of . What is the square of the product of the other (third) sides of and ?
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 ?
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
Raashan, Sylvia, and Ted play the following game. Each starts with . A bell rings every seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives to that player. What is the probability that after the bell has rung times, each player will have ? (For example, Raashan and Ted may each decide to give to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have , Sylvia will have , and Ted will have , and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their to, and the holdings will be the same at the end of the second round.)
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?
See also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2019 AMC 10A Problems |
Followed by 2020 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.