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Difference between revisions of "2023 AMC 10B Problems"

m (latex)
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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?
 
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]]
Line 46: Line 46:
 
==Problem 7==
 
==Problem 7==
  
−
Square <math>ABCD</math> is rotated <math>20</math> degrees clockwise about its center to obtain square <math>EFGH</math>, as shown below. What is the degree measure of <math>\angle{EAB}</math>?
+
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>?
  
−
[[Image:IMG%201031.jpeg]]
+
<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]]
Line 57: Line 71:
 
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>\text{(A)}\ 7 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 3</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]]
Line 64: Line 78:
 
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>?
 
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>\text{(A)}\ 674 \qquad \text{(B)}\ 1011 \qquad \text{(C)}\ 1010 \qquad \text{(D)}\ 2019 \qquad \text{(E)}\ 2017</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]]
Line 71: Line 85:
 
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?
 
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>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 6</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]]
Line 77: Line 91:
 
==Problem 11==
 
==Problem 11==
  
−
Suzanne went to the bank and withdrew <math>\$800</math>. The teller gave her this ammount 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?
+
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>
Line 86: Line 100:
 
When the roots of the polynomial  
 
When the roots of the polynomial  
  
−
<math>P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}</math>
+
<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? (2 different versions of this, some people had answer choices C and D flipped on the test)
+
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)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7</math>
+
<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
  
−
<math>| | x | - 1 | + | | y | - 1 | \le 1</math>?
+
<cmath>| | x | - 1 | + | | y | - 1 | \le 1?</cmath>
  
−
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12</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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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 <math>2</math> 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 <math>258</math> is an <math>upno</math> and <math>8620</math> is a <math>downno</math>. Let <math>\text{U}</math> equal the total
+
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 <math>\text{D}</math> equal the total number of <math>downnos</math>. What is <math>|\text{U}-\text{D}|</math>?
+
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 <math>\text{P}</math> has distinct edge lengths <math>a</math>, <math>b</math>, and <math>c</math>. The sum of the lengths of
+
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 <math>12</math> edges of <math>\text{P}</math> is <math>13</math>, the sum of the areas of all <math>6</math> faces of <math>\text{P}</math> is <math>\dfrac{11}{2}</math>, and the volume of <math>\text{P}</math> is <math>\dfrac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of <math>\text{P}</math>?
+
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>
Line 145: Line 159:
  
 
Suppose <math>a</math>, <math>b</math>, and <math>c</math> are positive integers such that
 
Suppose <math>a</math>, <math>b</math>, and <math>c</math> are positive integers such that
−
<math>\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}</math>.
+
<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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Four congruent semicircles are drawn on the surface of a sphere with radius <math>2</math>, 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 𝑛?
+
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>
−
 
−
[[Image:202310bQ20.jpeg]]
 
  
 
[[2023 AMC 10B Problems/Problem 20|Solution]]
 
[[2023 AMC 10B Problems/Problem 20|Solution]]
Line 192: Line 206:
 
==Problem 22==
 
==Problem 22==
 
How many distinct values of <math>x</math> satisfy
 
How many distinct values of <math>x</math> satisfy
−
<math>\lfloor{x}\rfloor^2-3x+2=0.</math>  
+
<math>\lfloor{x}\rfloor^2-3x+2=0,</math>  
 
where <math>\lfloor{x}\rfloor</math> denotes the largest integer less than or equal to <math>x</math>?
 
where <math>\lfloor{x}\rfloor</math> denotes the largest integer less than or equal to <math>x</math>?
  
Line 210: Line 224:
 
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) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16</math>
+
<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 12:42, 14 June 2024

2023 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

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 $\frac{1}{3}$ 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?

$\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

Solution

Problem 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

$\textbf{(A) }$46\qquad\textbf{(B) }$50\qquad\textbf{(C) }$48\qquad\textbf{(D) }$47\qquad\textbf{(E) }$49$

Solution

Problem 3

A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?

$\textbf{(A) }\frac{9}{25}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1 }{5}\qquad\textbf{(D) }\frac{25}{169}\qquad\textbf{(E) }\frac{4}{25}$

Solution

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?

$\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$

Solution

Problem 5

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?

$\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Solution

Problem 6

Let $L_1 = 1$, $L_2 = 3$, and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 1$. How many terms in the sequence $L_1, L_2, L_3, \cdots, L_{2023}$ are even?

$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$

Solution

Problem 7

Square $ABCD$ is rotated $20^{\circ}$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle{EAB}$?

[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]

$\textbf{(A) }24^{\circ}\qquad\textbf{(B) }35^{\circ}\qquad\textbf{(C) }30^{\circ}\qquad\textbf{(D) }32^{\circ}\qquad\textbf{(E) }20^{\circ}$

Solution

Problem 8

What is the units digit of $2022^{2023} + 2023^{2022}$?

$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 3$

Solution

Problem 9

The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$?

$\textbf{(A)}\ 674 \qquad \textbf{(B)}\ 1011 \qquad \textbf{(C)}\ 1010 \qquad \textbf{(D)}\ 2019 \qquad \textbf{(E)}\ 2017$

Solution

Problem 10

You are playing a game. A $2$ $\times$ $1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3$ $\times$ $3$ 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?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6$

Solution

Problem 11

Suzanne went to the bank and withdrew $$800$. The teller gave her this amount using $$20$ bills, $$50$ bills, and $$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?

$\textbf{(A) }45\qquad\textbf{(B) }21\qquad\textbf{(C) }36\qquad\textbf{(D) }28\qquad\textbf{(E) }32$

Solution

Problem 12

When the roots of the polynomial

\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]

are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?

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

Solution

Problem 13

What is the area of the region in the coordinate plane defined by

\[| | x | - 1 | + | | y | - 1 | \le 1?\]

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 12$

Solution

Problem 14

How many ordered pairs of integers $(m,n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$?

$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

Solution

Problem 15

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3!\cdot 4!\cdot 5! \dots 16!$ is a perfect square?

$\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

Solution

Problem 16

Define an $upno$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ to be a positive integer of $2$ 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 $U$ equal the total number of $upnos$ and let $D$ equal the total number of $downnos$. What is $|U-D|$?

$\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511$

Solution

Problem 17

A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the sum of the areas of all $6$ faces of $P$ is $\dfrac{11}{2}$, and the volume of $P$ is $\dfrac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $P$?

$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Solution

Problem 18

Suppose $a$, $b$, and $c$ are positive integers such that \[\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}.\]

Which of the following statements are necessarily true?

I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.

II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.

III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.

$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

Solution

Problem 19

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ 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?

$\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{4} \qquad \textbf{(D) } \frac{1}{10} \qquad \textbf{(E) } \frac{1}{9}$

Solution

Problem 20

Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is $n$?

202310bQ20.jpeg

$\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$

Solution

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?

$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

Solution

Problem 22

How many distinct values of $x$ satisfy $\lfloor{x}\rfloor^2-3x+2=0,$ where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$?

$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$

Solution

Problem 23

An arithmetic sequence of positive integers has $n\ge3$ terms, initial term $a$, and common difference $d>1$. Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$. The sum of the terms he wrote down was $222$. What is $a+d+n$?

$\textbf{(A) } 24 \qquad \textbf{(B) } 20 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 28 \qquad \textbf{(E) } 26$

Solution

Problem 24

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?

$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$

Solution

Problem 25

A regular pentagon with area $1+\sqrt5$ 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?

$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

Solution

See also

2023 AMC 10B (Problems • Answer Key • Resources)
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