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Difference between revisions of "2024 AMC 10B Problems"

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{{AMC10 Problems|year=2024|ab=B}}
 
{{AMC10 Problems|year=2024|ab=B}}
  
−
==Problem 1=
+
==Problem 1==
−
ur trying to cheat stop >:(  
+
 
 +
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
 +
 
 +
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
 +
 
 +
[[2024 AMC 10B Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
−
No seriously stop >:(
+
What is <math>10! - 7! \cdot 6!</math>
 +
 
 +
<math>\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720</math>
 +
 
 +
[[2024 AMC 10B Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
−
Why are you still here?
+
For how many integer values of <math>x</math> is <math>|2x| \leq 7 \pi</math>?
 +
 
 +
<math>\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textbf{(C) } 19 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 21</math>
 +
 
 +
[[2024 AMC 10B Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
−
Leave!
+
Balls numbered <math>1, 2, 3, \dots</math> are deposited in <math>5</math> bins, labeled <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> using the following procedure. Ball <math>1</math> is deposited in bin <math>A</math>, and balls <math>2</math> and <math>3</math> are deposited in bin <math>B</math>. The next <math>3</math> balls are deposited in bin <math>C</math>, the next <math>4</math> in bin <math>D</math>, and so on, cycling back to bin <math>A</math> after balls are deposited in bin <math>E</math>. (For example, balls numbered <math>22, 23, \dots, 28</math> are deposited in bin <math>B</math> at step <math>7</math> of this process.) In which bin is ball <math>2024</math> deposited?
 +
 
 +
<math>\textbf{(A) } A \qquad\textbf{(B) } B \qquad\textbf{(C) } C \qquad\textbf{(D) } D \qquad\textbf{(E) } E</math>
 +
 
 +
[[2024 AMC 10B Problems/Problem 4|Solution]]
 +
 
 
==Problem 5==
 
==Problem 5==
 +
In the following expression, Melanie changed some of the plus signs to minus signs:
 +
<cmath>1+3+5+7+...+97+99</cmath>
 +
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
 +
 +
<math>\textbf{(A) } 14 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 17 \qquad\textbf{(E) } 18</math>
 +
 +
[[2024 AMC 10B Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
 +
A rectangle has integer side lengths and an area of <math>2024</math>. What is the least possible perimeter of the rectangle?
 +
 +
<math>\textbf{(A) } 160 \qquad\textbf{(B) } 180 \qquad\textbf{(C) } 222 \qquad\textbf{(D) } 228 \qquad\textbf{(E) } 390</math>
 +
 +
[[2024 AMC 10B Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
 +
What is the remainder when <math>7^{2024}+7^{2025}+7^{2026}</math> is divided by <math>19</math>?
 +
 +
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 18</math>
 +
 +
[[2024 AMC 10B Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
 +
Let <math>N</math> be the product of all the positive integer divisors of <math>42</math>. What is the units digit
 +
of <math>N</math>?
 +
 +
<math>\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8</math>
 +
 +
[[2024 AMC 10B Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
Real numbers <math>a, b, </math> and <math>c</math> have arithmetic mean <math>0</math>. The arithmetic mean of <math>a^2, b^2, </math> and <math>c^2</math> is <math>10</math>. What is the arithmetic mean of <math>ab, ac, </math> and <math>bc</math>?
 +
 +
<math>\textbf{(A) } -5 \qquad\textbf{(B) } -\dfrac{10}{3} \qquad\textbf{(C) } -\dfrac{10}{9} \qquad\textbf{(D) } 0 \qquad\textbf{(E) } \dfrac{10}{9}</math>
 +
 +
[[2024 AMC 10B Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
Quadrilateral <math>ABCD</math> is a parallelogram, and <math>E</math> is the midpoint of the side <math>\overline{AD}</math>. Let <math>F</math> be the intersection of lines <math>EB</math> and <math>AC</math>. What is the ratio of the area of quadrilateral <math>CDEF</math> to the area of <math>\triangle CFB</math>?
 +
 +
<math>\textbf{(A) } 5:4 \qquad\textbf{(B) } 4:3 \qquad\textbf{(C) } 3:2 \qquad\textbf{(D) } 5:3 \qquad\textbf{(E) } 2:1</math>
 +
 +
[[2024 AMC 10B Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
In the figure below <math>WXYZ</math> is a rectangle with <math>WX=4</math> and <math>WZ=8</math>. Point <math>M</math> lies on <math>\overline{XY}</math>, point <math>A</math> lies on <math>\overline{YZ}</math>, and <math>\angle WMA</math> is a right angle. The areas of <math>\triangle WXM</math> and <math>\triangle W A Z</math> are equal. What is the area of <math>\triangle WMA</math>?
 +
 +
<asy>
 +
pair X = (0, 0);
 +
pair W = (0, 4);
 +
pair Y = (8, 0);
 +
pair Z = (8, 4);
 +
label("$X$", X, dir(180));
 +
label("$W$", W, dir(180));
 +
label("$Y$", Y, dir(0));
 +
label("$Z$", Z, dir(0));
 +
 +
draw(W--X--Y--Z--cycle);
 +
dot(X);
 +
dot(Y);
 +
dot(W);
 +
dot(Z);
 +
pair M = (2, 0);
 +
pair A = (8, 3);
 +
label("$A$", A, dir(0));
 +
dot(M);
 +
dot(A);
 +
draw(W--M--A--cycle);
 +
markscalefactor = 0.05;
 +
draw(rightanglemark(W, M, A));
 +
label("$M$", M, dir(-90));
 +
</asy>
 +
 +
<math>
 +
\textbf{(A) }13 \qquad
 +
\textbf{(B) }14 \qquad
 +
\textbf{(C) }15 \qquad
 +
\textbf{(D) }16 \qquad
 +
\textbf{(E) }17 \qquad
 +
</math>
 +
 +
[[2024 AMC 10B Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
A group of <math>100</math> students from different countries meet at a mathematics competition.
 +
Each student speaks the same number of languages, and, for every pair of
 +
students <math>A</math> and <math>B</math>, student <math>A</math> speaks some language that student <math>B</math> does not speak,
 +
and student <math>B</math> speaks some language that student <math>A</math> does not speak. What is the
 +
least possible total number of languages spoken by all the students?
 +
 +
<math>\textbf{(A) } 9 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 51 \qquad\textbf{(E) } 100</math>
 +
 +
[[2024 AMC 10B Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
Positive integers <math>x</math> and <math>y</math> satisfy the equation <math>\sqrt{x} + \sqrt{y} = \sqrt{1183}</math>. What is the minimum possible value of <math>x+y</math>?
 +
 +
<math>\textbf{(A) } 585 \qquad\textbf{(B) } 595 \qquad\textbf{(C) } 623 \qquad\textbf{(D) } 700 \qquad\textbf{(E) } 791</math>
 +
 +
[[2024 AMC 10B Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
A dartboard is the region <math>B</math> in the coordinate plane consisting of points <math>(x,y)</math> such
 +
that <math>|x| + |y| \le 8</math> . A target <math>T</math> is the region where <math>(x^2 + y^2 - 25)^2 \le 49.</math> A dart is
 +
thrown and lands at a random point in <math>B</math>. The probability that the dart lands in <math>T</math> can be
 +
expressed as <math>\frac{m}{n} \cdot \pi,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What
 +
is <math>m + n?</math>
 +
 +
<math>\textbf{(A) } 39 \qquad\textbf{(B) } 71 \qquad\textbf{(C) } 73 \qquad\textbf{(D) } 75 \qquad\textbf{(E) } 135</math>
 +
 +
[[2024 AMC 10B Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
A list of <math>9</math> real numbers consists of <math>1</math>, <math>2.2</math>, <math>3.2</math>, <math>5.2</math>, <math>6.2</math>, and <math>7</math>, as well as <math>x</math>, <math>y</math> , and <math>z</math> with <math>x</math> <math>\le</math> <math>y</math> <math>\le</math> <math>z</math>. The range of the list is <math>7</math>, and the mean and the median are both positive integers. How many ordered triples (<math>x</math>, <math>y</math>, <math>z</math>) are possible?
 +
 +
<math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } \text{infinitely many}</math>
 +
 +
[[2024 AMC 10B Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
Jerry likes to play with numbers. One day, he wrote all the integers from <math>1</math> to <math>2024</math> on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase <math>1</math>, <math>2</math>, <math>3</math>, and <math>5</math>, and then write either <math>11</math>, their sum, or <math>30</math>, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
 +
 +
<math>\textbf{(A) } 1010 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1012 \qquad \textbf{(D) } 1013 \qquad \textbf{(E) } 1014</math>
 +
 +
[[2024 AMC 10B Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
In a race among 5 snails, there is at most one tie, but that tie can involve any number
 +
of snails. For example, the result of the race might be that Dazzler is first; Abby,
 +
Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results
 +
of the race are possible?
 +
 +
<math>\textbf{(A) } 180 \qquad\textbf{(B) } 361 \qquad\textbf{(C) } 420 \qquad\textbf{(D) } 431 \qquad\textbf{(E) } 720</math>
 +
 +
[[2024 AMC 10B Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
 +
How many different remainders can result when the <math>100</math>th power of an integer is
 +
divided by <math>125</math>?
 +
 +
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 125</math>
 +
 +
[[2024 AMC 10B Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
 +
In the following table, each question mark is to be replaced by "Possible" or "Not
 +
Possible" to indicate whether a nonvertical line with the given slope can contain the
 +
given number of lattice points (points both of whose coordinates are integers). How
 +
many of the 12 entries will be "Possible"?
 +
 +
[[File:AMC10B2024 P19.png]]
 +
<math>\textbf{(A) } 4 \qquad    \textbf{(B) } 5 \qquad \textbf{(C) } 6    \qquad \textbf{(D) } 7      \qquad \textbf{(E) } 9</math>
 +
 +
[[2024 AMC 10B Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
 +
Three different pairs of shoes are placed in a row so that no left shoe is next to a
 +
right shoe from a different pair. In how many ways can these six shoes be lined up?
 +
 +
<math>\textbf{(A) } 60 \qquad\textbf{(B) } 72 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 108 \qquad\textbf{(E) } 120</math>
 +
 +
[[2024 AMC 10B Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
 +
Two straight pipes (circular cylinders), with radii <math>1</math> and <math>\frac{1}{4}</math>, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
 +
 +
<asy>
 +
size(6cm);
 +
draw(circle((0,1),1), linewidth(1.2));
 +
draw((-1,0)--(1.25,0), linewidth(1.2));
 +
draw(circle((1,1/4),1/4), linewidth(1.2));
 +
</asy>
 +
 +
<math>\textbf{(A)}~\frac{1}{9}
 +
\qquad\textbf{(B)}~1
 +
\qquad\textbf{(C)}~\frac{10}{9}
 +
\qquad\textbf{(D)}~\frac{11}{9}
 +
\qquad\textbf{(E)}~\frac{19}{9}</math>
 +
 +
[[2024 AMC 10B Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
 +
A group of <math>16</math> people will be partitioned into <math>4</math> indistinguishable <math>4</math>-person
 +
committees. Each committee will have one chairperson and one secretary. The
 +
number of different ways to make these assignments can be written
 +
as <math>{3^r}M</math>, where <math>r</math> and <math>M</math> are positive integers and <math>M</math> is not divisible by <math>3</math>. What is <math>r</math>?
 +
 +
<math>\textbf{(A) } 5 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 9</math>
 +
 +
[[2024 AMC 10B Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
The Fibonacci numbers are defined by <math>F_1 = 1, F_2 = 1,</math> and <math>F_n = F_{n-1} + F_{n-2}</math> for <math>n \geq 3.</math> What is <cmath>{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?</cmath>
 +
<math>\textbf{(A) } 318 \qquad\textbf{(B) } 319 \qquad\textbf{(C) } 320 \qquad\textbf{(D) } 321 \qquad\textbf{(E) } 322</math>
 +
 +
[[2024 AMC 10B Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
Let
 +
<cmath>P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}</cmath>
 +
How many of the values <math>P(2022)</math>, <math>P(2023)</math>, <math>P(2024)</math>, and <math>P(2025)</math> are integers?
 +
 +
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
 +
 +
[[2024 AMC 10B Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
−
Hey! just get lost!
+
Each of <math>27</math> bricks (right rectangular prisms) has dimensions <math>a \times b \times c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are pairwise relatively prime positive integers. These bricks are arranged to form a <math>3 \times 3 \times 3</math> block, as shown on the left below. A <math>28</math>th brick with the same dimensions is introduced, and these bricks are reconfigured into a <math>2 \times 2 \times 7</math> block, shown on the right. The new block is <math>1</math> unit taller, <math>1</math> unit wider, and <math>1</math> unit deeper than the old one. What is <math>a + b + c</math>?
 +
 
 +
[[File:AMC10B2024 P25.png]]
 +
 
 +
<math>\textbf{(A) } 88 \qquad\textbf{(B) } 89 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 91 \qquad\textbf{(E) } 92</math>
 +
 
 +
[[2024 AMC 10B Problems/Problem 25|Solution]]
  
 
==See also==
 
==See also==
−
{{AMC10 box|year=2024|ab=A|before=[[2024 AMC 10A Problems]]|after=[[2025 AMC 10A Problems]]}}
+
{{AMC10 box|year=2024|ab=B|before=[[2024 AMC 10A Problems]]|after=[[2025 AMC 10A Problems]]}}
 
* [[AMC 10]]
 
* [[AMC 10]]
 
* [[AMC 10 Problems and Solutions]]
 
* [[AMC 10 Problems and Solutions]]

Latest revision as of 11:46, 17 November 2024

2024 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

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

$\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025$

Solution

Problem 2

What is $10! - 7! \cdot 6!$

$\textbf{(A) } -120 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 600 \qquad\textbf{(E) } 720$

Solution

Problem 3

For how many integer values of $x$ is $|2x| \leq 7 \pi$?

$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textbf{(C) } 19 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 21$

Solution

Problem 4

Balls numbered $1, 2, 3, \dots$ are deposited in $5$ bins, labeled $A$, $B$, $C$, $D$, and $E$ using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposited in bin $B$. The next $3$ balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, balls numbered $22, 23, \dots, 28$ are deposited in bin $B$ at step $7$ of this process.) In which bin is ball $2024$ deposited?

$\textbf{(A) } A \qquad\textbf{(B) } B \qquad\textbf{(C) } C \qquad\textbf{(D) } D \qquad\textbf{(E) } E$

Solution

Problem 5

In the following expression, Melanie changed some of the plus signs to minus signs: \[1+3+5+7+...+97+99\] When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

$\textbf{(A) } 14 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 17 \qquad\textbf{(E) } 18$

Solution

Problem 6

A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle?

$\textbf{(A) } 160 \qquad\textbf{(B) } 180 \qquad\textbf{(C) } 222 \qquad\textbf{(D) } 228 \qquad\textbf{(E) } 390$

Solution

Problem 7

What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 18$

Solution

Problem 8

Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?

$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

Solution

Problem 9

Real numbers $a, b,$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2,$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac,$ and $bc$?

$\textbf{(A) } -5 \qquad\textbf{(B) } -\dfrac{10}{3} \qquad\textbf{(C) } -\dfrac{10}{9} \qquad\textbf{(D) } 0 \qquad\textbf{(E) } \dfrac{10}{9}$

Solution

Problem 10

Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of $\triangle CFB$?

$\textbf{(A) } 5:4 \qquad\textbf{(B) } 4:3 \qquad\textbf{(C) } 3:2 \qquad\textbf{(D) } 5:3 \qquad\textbf{(E) } 2:1$

Solution

Problem 11

In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies on $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle W A Z$ are equal. What is the area of $\triangle WMA$?

[asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy]

$\textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad$

Solution

Problem 12

A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?

$\textbf{(A) } 9 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 51 \qquad\textbf{(E) } 100$

Solution

Problem 13

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$. What is the minimum possible value of $x+y$?

$\textbf{(A) } 585 \qquad\textbf{(B) } 595 \qquad\textbf{(C) } 623 \qquad\textbf{(D) } 700 \qquad\textbf{(E) } 791$

Solution

Problem 14

A dartboard is the region $B$ in the coordinate plane consisting of points $(x,y)$ such that $|x| + |y| \le 8$ . A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49.$ A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

$\textbf{(A) } 39 \qquad\textbf{(B) } 71 \qquad\textbf{(C) } 73 \qquad\textbf{(D) } 75 \qquad\textbf{(E) } 135$

Solution

Problem 15

A list of $9$ real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, and $7$, as well as $x$, $y$ , and $z$ with $x$ $\le$ $y$ $\le$ $z$. The range of the list is $7$, and the mean and the median are both positive integers. How many ordered triples ($x$, $y$, $z$) are possible?

$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } \text{infinitely many}$

Solution

Problem 16

Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase $1$, $2$, $3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?

$\textbf{(A) } 1010 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1012 \qquad \textbf{(D) } 1013 \qquad \textbf{(E) } 1014$

Solution

Problem 17

In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?

$\textbf{(A) } 180 \qquad\textbf{(B) } 361 \qquad\textbf{(C) } 420 \qquad\textbf{(D) } 431 \qquad\textbf{(E) } 720$

Solution

Problem 18

How many different remainders can result when the $100$th power of an integer is divided by $125$?

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 125$

Solution

Problem 19

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?

AMC10B2024 P19.png $\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

Solution

Problem 20

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?

$\textbf{(A) } 60 \qquad\textbf{(B) } 72 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 108 \qquad\textbf{(E) } 120$

Solution

Problem 21

Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?

[asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy]

$\textbf{(A)}~\frac{1}{9} \qquad\textbf{(B)}~1 \qquad\textbf{(C)}~\frac{10}{9} \qquad\textbf{(D)}~\frac{11}{9} \qquad\textbf{(E)}~\frac{19}{9}$

Solution

Problem 22

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as ${3^r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?

$\textbf{(A) } 5 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 9$

Solution

Problem 23

The Fibonacci numbers are defined by $F_1 = 1, F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3.$ What is \[{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?\] $\textbf{(A) } 318 \qquad\textbf{(B) } 319 \qquad\textbf{(C) } 320 \qquad\textbf{(D) } 321 \qquad\textbf{(E) } 322$

Solution

Problem 24

Let \[P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}\] How many of the values $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers?

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Problem 25

Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$th brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$?

AMC10B2024 P25.png

$\textbf{(A) } 88 \qquad\textbf{(B) } 89 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 91 \qquad\textbf{(E) } 92$

Solution

See also

2024 AMC 10B (Problems • Answer Key • Resources)
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2024 AMC 10A Problems 		Followed by
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All AMC 10 Problems and Solutions

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