Difference between revisions of "2023 AMC 10B Problems"
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==Problem 4== | ==Problem 4== | ||
− | Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint? | + | Jackson's paintbrush makes a narrow strip with a width of <math>6.5</math> millimeters. Jackson has enough paint to make a strip <math>25</math> meters long. How many square centimeters of paper could Jackson cover with paint? |
− | <math>\textbf{(A) }162,500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1,625\qquad\textbf{(D) }1,625,000\qquad\textbf{(E) }16,250</math> | + | <math>\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250</math> |
[[2023 AMC 10B Problems/Problem 4|Solution]] | [[2023 AMC 10B Problems/Problem 4|Solution]] | ||
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Maddy and Lara see a list of numbers written on a blackboard. Maddy adds <math>3</math> to each number in the list and finds that the sum of her new numbers is <math>45</math>. Lara multiplies each number in the list by <math>3</math> and finds that the sum of her new numbers is also <math>45</math>. How many numbers are written on the blackboard? | Maddy and Lara see a list of numbers written on a blackboard. Maddy adds <math>3</math> to each number in the list and finds that the sum of her new numbers is <math>45</math>. Lara multiplies each number in the list by <math>3</math> and finds that the sum of her new numbers is also <math>45</math>. How many numbers are written on the blackboard? | ||
− | <math>\textbf{(A) } | + | <math>\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math> |
[[2023 AMC 10B Problems/Problem 5|Solution]] | [[2023 AMC 10B Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | Let <math> | + | Let <math>L_1 = 1</math>, <math>L_2 = 3</math>, and <math>L_{n+2} = L_{n+1}+L_n</math> for <math>n \geq 1</math>. How many terms in the sequence <math>L_1, L_2, L_3, \cdots, L_{2023}</math> are even? |
− | <math>\textbf{(A) }673\qquad\textbf{(B) } | + | <math>\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674</math> |
[[2023 AMC 10B Problems/Problem 6|Solution]] | [[2023 AMC 10B Problems/Problem 6|Solution]] | ||
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==Problem 7== | ==Problem 7== | ||
− | Square ABCD is rotated 20 | + | Square <math>ABCD</math> is rotated <math>20^{\circ}</math> clockwise about its center to obtain square <math>EFGH</math>, as shown below. What is the degree measure of <math>\angle{EAB}</math>? |
− | + | <asy> | |
+ | size(170); | ||
+ | defaultpen(linewidth(0.6)); | ||
+ | real r = 25; | ||
+ | draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle); | ||
+ | draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle); | ||
+ | label("$A$",dir(135),NW); | ||
+ | label("$B$",dir(45),NE); | ||
+ | label("$C$",dir(315),SE); | ||
+ | label("$D$",dir(225),SW); | ||
+ | label("$E$",dir(135-r),N); | ||
+ | label("$F$",dir(45-r),E); | ||
+ | label("$G$",dir(315-r),S); | ||
+ | label("$H$",dir(225-r),W); | ||
+ | </asy> | ||
− | <math>\textbf{(A) }24\qquad\textbf{(B) }35\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }20</math> | + | <math>\textbf{(A) }24^{\circ}\qquad\textbf{(B) }35^{\circ}\qquad\textbf{(C) }30^{\circ}\qquad\textbf{(D) }32^{\circ}\qquad\textbf{(E) }20^{\circ}</math> |
[[2023 AMC 10B Problems/Problem 7|Solution]] | [[2023 AMC 10B Problems/Problem 7|Solution]] | ||
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What is the units digit of <math>2022^{2023} + 2023^{2022}</math>? | What is the units digit of <math>2022^{2023} + 2023^{2022}</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 3</math> |
[[2023 AMC 10B Problems/Problem 8|Solution]] | [[2023 AMC 10B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | The numbers 16 and 25 are a pair of consecutive | + | The numbers <math>16</math> and <math>25</math> are a pair of consecutive positive squares whose difference is <math>9</math>. How many pairs of consecutive positive perfect squares have a difference of less than or equal to <math>2023</math>? |
− | <math>\ | + | <math>\textbf{(A)}\ 674 \qquad \textbf{(B)}\ 1011 \qquad \textbf{(C)}\ 1010 \qquad \textbf{(D)}\ 2019 \qquad \textbf{(E)}\ 2017</math> |
[[2023 AMC 10B Problems/Problem 9|Solution]] | [[2023 AMC 10B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | You are playing a game. A 2 | + | You are playing a game. A <math>2</math> <math>\times</math> <math>1</math> rectangle covers two adjacent squares (oriented either horizontally or vertically) of a <math>3</math> <math>\times</math> <math>3</math> grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle? |
− | <math>\ | + | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6</math> |
[[2023 AMC 10B Problems/Problem 10|Solution]] | [[2023 AMC 10B Problems/Problem 10|Solution]] | ||
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==Problem 11== | ==Problem 11== | ||
− | Suzanne went to the bank and withdrew 800 | + | Suzanne went to the bank and withdrew <math>\$800</math>. The teller gave her this amount using <math>\$20</math> bills, <math>\$50</math> bills, and <math>\$100</math> bills, with at least one of each denomination. How many different collections of bills could Suzanne have received? |
<math>\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32</math> | <math>\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32</math> | ||
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When the roots of the polynomial | When the roots of the polynomial | ||
− | < | + | <cmath>P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}</cmath> |
− | are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is <math>P(x)</math> positive? | + | are removed from the number line, what remains is the union of <math>11</math> disjoint open intervals. On how many of these intervals is <math>P(x)</math> positive? |
− | <math>\textbf{(A)}~3\qquad\textbf{(B)}~ | + | <math>\textbf{(A)}~3\qquad\textbf{(B)}~7\qquad\textbf{(C)}~6\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5</math> |
[[2023 AMC 10B Problems/Problem 12|Solution]] | [[2023 AMC 10B Problems/Problem 12|Solution]] | ||
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What is the area of the region in the coordinate plane defined by | What is the area of the region in the coordinate plane defined by | ||
− | < | + | <cmath>| | x | - 1 | + | | y | - 1 | \le 1?</cmath> |
− | <math>\ | + | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 12</math> |
[[2023 AMC 10B Problems/Problem 13|Solution]] | [[2023 AMC 10B Problems/Problem 13|Solution]] | ||
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==Problem 16== | ==Problem 16== | ||
− | Define an <math>upno</math> to be a positive integer of 2 or more digits where the digits are strictly | + | Define an <math>upno</math> to be a positive integer of <math>2</math> or more digits where the digits are strictly |
increasing moving left to right. Similarly, define a <math>downno</math> to be a positive integer | increasing moving left to right. Similarly, define a <math>downno</math> to be a positive integer | ||
− | of 2 or more digits where the digits are strictly decreasing moving left to right. For | + | of <math>2</math> or more digits where the digits are strictly decreasing moving left to right. For |
− | instance, the number 258 is an upno and 8620 is a downno. Let | + | instance, the number <math>258</math> is an upno and <math>8620</math> is a downno. Let <math>U</math> equal the total |
− | number of <math>upnos</math> and let | + | number of <math>upnos</math> and let <math>D</math> equal the total number of <math>downnos</math>. What is <math>|U-D|</math>? |
<math>\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511</math> | <math>\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511</math> | ||
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==Problem 17== | ==Problem 17== | ||
− | A rectangular box | + | A rectangular box <math>P</math> has distinct edge lengths <math>a</math>, <math>b</math>, and <math>c</math>. The sum of the lengths of |
− | all 12 edges of | + | all <math>12</math> edges of <math>P</math> is <math>13</math>, the sum of the areas of all <math>6</math> faces of <math>P</math> is <math>\dfrac{11}{2}</math>, and the volume of <math>P</math> is <math>\dfrac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of <math>P</math>? |
<math>\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}</math> | <math>\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}</math> | ||
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− | Suppose | + | Suppose <math>a</math>, <math>b</math>, and <math>c</math> are positive integers such that |
− | < | + | <cmath>\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}.</cmath> |
Which of the following statements are necessarily true? | Which of the following statements are necessarily true? | ||
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==Problem 19== | ==Problem 19== | ||
Sonya the frog chooses a point uniformly at random lying within the square | Sonya the frog chooses a point uniformly at random lying within the square | ||
− | <math>[0, 6] | + | <math>[0, 6]</math> <math>\times</math> <math>[0, 6]</math> in the coordinate plane and hops to that point. She then randomly |
chooses a distance uniformly at random from <math>[0, 1]</math> and a direction uniformly at | chooses a distance uniformly at random from <math>[0, 1]</math> and a direction uniformly at | ||
− | random from {north, south east, west}. All | + | random from {north, south, east, west}. All her choices are independent. She now |
hops the distance in the chosen direction. What is the probability that she lands | hops the distance in the chosen direction. What is the probability that she lands | ||
outside the square? | outside the square? | ||
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==Problem 20== | ==Problem 20== | ||
− | Four congruent semicircles are drawn on the surface of a sphere with radius 2, as | + | Four congruent semicircles are drawn on the surface of a sphere with radius <math>2</math>, as |
shown, creating a close curve that divides the surface into two congruent regions. | shown, creating a close curve that divides the surface into two congruent regions. | ||
− | The length of the curve is <math>\pi\sqrt{n}</math>. What is | + | The length of the curve is <math>\pi\sqrt{n}</math>. What is <math>n</math>? |
+ | |||
+ | [[Image:202310bQ20.jpeg|center]] | ||
<math>\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27</math> | <math>\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27</math> | ||
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==Problem 21== | ==Problem 21== | ||
− | Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? | + | Each of <math>2023</math> balls is randomly placed into one of <math>3</math> bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? |
<math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}</math> | <math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}</math> | ||
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==Problem 22== | ==Problem 22== | ||
− | How many distinct values of | + | How many distinct values of <math>x</math> satisfy |
− | <math>\lfloor{x}\rfloor^2-3x+2=0 | + | <math>\lfloor{x}\rfloor^2-3x+2=0,</math> |
− | where <math>\lfloor{x}\rfloor</math> denotes the largest integer less than or equal to | + | where <math>\lfloor{x}\rfloor</math> denotes the largest integer less than or equal to <math>x</math>? |
<math>\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0</math> | <math>\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0</math> | ||
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What is the perimeter of the boundary of the region consisting of all points which can be expressed as <math>(2u-3w, v+4w)</math> with <math>0\le u\le1</math>, <math>0\le v\le1,</math> and <math>0\le w\le1</math>? | What is the perimeter of the boundary of the region consisting of all points which can be expressed as <math>(2u-3w, v+4w)</math> with <math>0\le u\le1</math>, <math>0\le v\le1,</math> and <math>0\le w\le1</math>? | ||
− | <math>\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } | + | <math>\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16</math> |
[[2023 AMC 10B Problems/Problem 24|Solution]] | [[2023 AMC 10B Problems/Problem 24|Solution]] |
Latest revision as of 13:42, 14 June 2024
2023 AMC 10B (Answer Key) Printable versions: • 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
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
Problem 2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by on every pair of shoes. Carlos also knew that he had to pay a
sales tax on the discounted price. He had
dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
Problem 3
A right triangle is inscribed in circle
, and a
right triangle is inscribed in circle
. What is the ratio of the area of circle
to the area of circle
?
Problem 4
Jackson's paintbrush makes a narrow strip with a width of millimeters. Jackson has enough paint to make a strip
meters long. How many square centimeters of paper could Jackson cover with paint?
Problem 5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds to each number in the list and finds that the sum of her new numbers is
. Lara multiplies each number in the list by
and finds that the sum of her new numbers is also
. How many numbers are written on the blackboard?
Problem 6
Let ,
, and
for
. How many terms in the sequence
are even?
Problem 7
Square is rotated
clockwise about its center to obtain square
, as shown below. What is the degree measure of
?
Problem 8
What is the units digit of ?
Problem 9
The numbers and
are a pair of consecutive positive squares whose difference is
. How many pairs of consecutive positive perfect squares have a difference of less than or equal to
?
Problem 10
You are playing a game. A
rectangle covers two adjacent squares (oriented either horizontally or vertically) of a
grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
Problem 11
Suzanne went to the bank and withdrew . The teller gave her this amount using
bills,
bills, and
bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
Problem 12
When the roots of the polynomial
are removed from the number line, what remains is the union of disjoint open intervals. On how many of these intervals is
positive?
Problem 13
What is the area of the region in the coordinate plane defined by
Problem 14
How many ordered pairs of integers satisfy the equation
?
Problem 15
What is the least positive integer such that
is a perfect square?
Problem 16
Define an to be a positive integer of
or more digits where the digits are strictly
increasing moving left to right. Similarly, define a
to be a positive integer
of
or more digits where the digits are strictly decreasing moving left to right. For
instance, the number
is an upno and
is a downno. Let
equal the total
number of
and let
equal the total number of
. What is
?
Problem 17
A rectangular box has distinct edge lengths
,
, and
. The sum of the lengths of
all
edges of
is
, the sum of the areas of all
faces of
is
, and the volume of
is
. What is the length of the longest interior diagonal connecting two vertices of
?
Problem 18
Suppose ,
, and
are positive integers such that
Which of the following statements are necessarily true?
I. If or
or both, then
.
II. If , then
or
or both.
III. if and only if
.
Problem 19
Sonya the frog chooses a point uniformly at random lying within the square
in the coordinate plane and hops to that point. She then randomly
chooses a distance uniformly at random from
and a direction uniformly at
random from {north, south, east, west}. All her choices are independent. She now
hops the distance in the chosen direction. What is the probability that she lands
outside the square?
Problem 20
Four congruent semicircles are drawn on the surface of a sphere with radius , as
shown, creating a close curve that divides the surface into two congruent regions.
The length of the curve is
. What is
?
Problem 21
Each of balls is randomly placed into one of
bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
Problem 22
How many distinct values of satisfy
where
denotes the largest integer less than or equal to
?
Problem 23
An arithmetic sequence of positive integers has terms, initial term
, and common difference
. Carl wrote down all the terms in this sequence correctly except for one term, which was off by
. The sum of the terms he wrote down was
. What is
?
Problem 24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as with
,
and
?
Problem 25
A regular pentagon with area is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2023 AMC 10A Problems |
Followed by 2024 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.