Difference between revisions of "2024 AMC 12A Problems"

Revision as of 17:12, 8 November 2024

2024 AMC 12A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. 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

What is the value of $9901\cdot101-99\cdot10101?$

$\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020$

Solution

Problem 2

A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

$\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264$

Solution

Problem 3

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Solution

Problem 4

What is the least value of $n$ such that $n!$ is a multiple of $2024$ ?

$\textbf{(A) }11 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22 \qquad \textbf{(D) }23 \qquad \textbf{(E) }253 \qquad$

Solution

Problem 5

A data set containing $20$ numbers, some of which are $6$ , has mean $45$ . When all the 6s are removed, the data set has mean $66$ . How many 6s were in the original data set?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Solution

Problem 6

The product of three integers is $60$ . What is the least possible positive sum of the three integers?

$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

Solution

Problem 7

In $\Delta ABC$ , $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$ . Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$ . What is the length of the vector sum $\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?$ $\textbf{(A) }1011 \qquad \textbf{(B) }1012 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad$

Solution

Problem 8

How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$ ? $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad$

Solution

Problem 9

Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$ ?

$\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad$

Solution

Problem 10

Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$ , what is $\beta$ ?

$\textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad$

Solution

Problem 11

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?

$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solution

Problem 12

The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$

$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$

Solution

Problem 13

The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?

$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$

Solution

Problem 14

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$ $\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}$ $\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$

Solution

Problem 15

The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of $(p^2 + 4)(q^2 + 4)(r^2 + 4)?$ $\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144$

Solution

Problem 16

A set of $12$ tokens ---- $3$ red, $2$ white, $1$ blue, and $6$ black ---- is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? $\textbf{(A) }387 \qquad \textbf{(B) }388 \qquad \textbf{(C) }389 \qquad \textbf{(D) }390 \qquad \textbf{(E) }391 \qquad$

Solution

Problem 17

Integers $a$ , $b$ , and $c$ satisfy $ab + c = 100$ , $bc + a = 87$ , and $ca + b = 60$ . What is $ab + bc + ca$ ?

$\textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad$

Solution

Problem 18

On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$ , an identical card is placed so that two of their diagonals line up, as shown ( $\overline{AC}$ , in this case).

$[asy] defaultpen(fontsize(12)+0.85); size(150); real h=2.25; pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B; pair L=extension(A,Dp,B,C),R=extension(Bp,C,A,D); draw(L--B--A--Dp--C--Bp--A); draw(C--D--R); draw(L--C^^R--A,dashed+0.6); draw(A--C,black+0.6); dot("$C$",C,2*dir(C-R)); dot("$A$",A,1.5*dir(A-L)); dot("$B$",B,dir(B-R)); [/asy]$

Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled $B$ in the figure?

$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.$

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

@@ Line 18: / Line 18: @@
 ==Problem 3==
+The number <math>2024</math> is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
+<math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math>
 [[2024 AMC 12A Problems/Problem 3|Solution]]
 ==Problem 4==
+What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
+<math>
+\textbf{(A) }11 \qquad
+\textbf{(B) }21 \qquad
+\textbf{(C) }22 \qquad
+\textbf{(D) }23 \qquad
+\textbf{(E) }253 \qquad
+</math>
 [[2024 AMC 12A Problems/Problem 4|Solution]]
 ==Problem 5==
+A data set containing <math>20</math> numbers, some of which are <math>6</math>, has mean <math>45</math>. When all the 6s are removed, the data set has mean <math>66</math>. How many 6s were in the original data set?
+<math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math>
 [[2024 AMC 12A Problems/Problem 5|Solution]]
 ==Problem 6==
+The product of three integers is <math>60</math>. What is the least possible positive sum of the three integers?
+<math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13</math>
 [[2024 AMC 12A Problems/Problem 6|Solution]]
 ==Problem 7==
+In <math>\Delta ABC</math>, <math>\angle ABC = 90^\circ</math> and <math>BA = BC = \sqrt{2}</math>. Points <math>P_1, P_2, \dots, P_{2024}</math> lie on hypotenuse <math>\overline{AC}</math> so that <math>AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C</math>. What is the length of the vector sum
+<cmath> \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? </cmath>
+<math>
+\textbf{(A) }1011 \qquad
+\textbf{(B) }1012 \qquad
+\textbf{(C) }2023 \qquad
+\textbf{(D) }2024 \qquad
+\textbf{(E) }2025 \qquad
+</math>
 [[2024 AMC 12A Problems/Problem 7|Solution]]
 ==Problem 8==
+How many angles <math>\theta</math> with <math>0\le\theta\le2\pi</math> satisfy <math>\log(\sin(3\theta))+\log(\cos(2\theta))=0</math>?  <math> \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad </math>
 [[2024 AMC 12A Problems/Problem 8|Solution]]
 ==Problem 9==
+Let <math>M</math> be the greatest integer such that both <math>M + 1213</math> and <math>M + 3773</math> are perfect squares. What is the units digit of <math>M</math>?
+<math>
+\textbf{(A) }1 \qquad
+\textbf{(B) }2 \qquad
+\textbf{(C) }3 \qquad
+\textbf{(D) }6 \qquad
+\textbf{(E) }8 \qquad
+</math>
 [[2024 AMC 12A Problems/Problem 9|Solution]]
 ==Problem 10==
+Let <math>\alpha</math> be the radian measure of the smallest angle in a <math>3{-}4{-}5</math> right triangle. Let <math>\beta</math> be the radian measure of the smallest angle in a <math>7{-}24{-}25</math> right triangle. In terms of <math>\alpha</math>, what is <math>\beta</math>?
+<math>
+\textbf{(A) }\frac{\alpha}{3}\qquad
+\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
+\textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad
+\textbf{(D) }\frac{\alpha}{2}\qquad
+\textbf{(E) }\pi - 4\alpha\qquad
+</math>
 [[2024 AMC 12A Problems/Problem 10|Solution]]
 ==Problem 11==
+There are exactly <math>K</math> positive integers <math>b</math> with <math>5 \leq b \leq 2024</math> such that the base-<math>b</math> integer <math>2024_b</math> is divisible by <math>16</math> (where <math>16</math> is in base ten). What is the sum of the digits of <math>K</math>?
+<math>\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21</math>
 [[2024 AMC 12A Problems/Problem 11|Solution]]
 ==Problem 12==
+The first three terms of a geometric sequence are the integers <math>a,\,720,</math> and <math>b,</math> where <math>a<720<b.</math> What is the sum of the digits of the least possible value of <math>b?</math>
+<math>\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21</math>
 [[2024 AMC 12A Problems/Problem 12|Solution]]
 ==Problem 13==
+The graph of <math>y=e^{x+1}+e^{-x}-2</math> has an axis of symmetry. What is the reflection of the point <math>(-1,\tfrac{1}{2})</math> over this axis?
+<math>\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)</math>
 [[2024 AMC 12A Problems/Problem 13|Solution]]
 ==Problem 14==
+The numbers, in order, of each row and the numbers, in order, of each column of a <math>5 \times 5</math> array of integers form an arithmetic progression of length <math>5{.}</math> The numbers in positions <math>(5, 5), \,(2,4),\,(4,3),</math> and <math>(3, 1)</math> are <math>0, 48, 16,</math> and <math>12{,}</math> respectively. What number is in position <math>(1, 2)?</math>
+<cmath> \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}</cmath>
+<math>\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39</math>
 [[2024 AMC 12A Problems/Problem 14|Solution]]
 ==Problem 15==
+The roots of <math>x^3 + 2x^2 - x + 3</math> are <math>p, q,</math> and <math>r.</math> What is the value of <cmath>(p^2 + 4)(q^2 + 4)(r^2 + 4)?</cmath>
+<math>\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144</math>
 [[2024 AMC 12A Problems/Problem 15|Solution]]
 ==Problem 16==
+A set of <math>12</math> tokens ---- <math>3</math> red, <math>2</math> white, <math>1</math> blue, and <math>6</math> black ---- is to be distributed at random to <math>3</math> game players, <math>4</math> tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>?
+<math>
+\textbf{(A) }387 \qquad
+\textbf{(B) }388 \qquad
+\textbf{(C) }389 \qquad
+\textbf{(D) }390 \qquad
+\textbf{(E) }391 \qquad
+</math>
 [[2024 AMC 12A Problems/Problem 16|Solution]]
 ==Problem 17==
+Integers <math>a</math>, <math>b</math>, and <math>c</math> satisfy <math>ab + c = 100</math>, <math>bc + a = 87</math>, and <math>ca + b = 60</math>. What is <math>ab + bc + ca</math>?
+<math>
+\textbf{(A) }212 \qquad
+\textbf{(B) }247 \qquad
+\textbf{(C) }258 \qquad
+\textbf{(D) }276 \qquad
+\textbf{(E) }284 \qquad
+</math>
 [[2024 AMC 12A Problems/Problem 17|Solution]]
 ==Problem 18==
+On top of a rectangular card with sides of length <math>1</math> and <math>2+\sqrt{3}</math>, an identical card is placed so that two of their diagonals line up, as shown (<math>\overline{AC}</math>, in this case).
+<asy>
+defaultpen(fontsize(12)+0.85); size(150);
+real h=2.25;
+pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B;
+pair L=extension(A,Dp,B,C),R=extension(Bp,C,A,D);
+draw(L--B--A--Dp--C--Bp--A);
+draw(C--D--R);
+draw(L--C^^R--A,dashed+0.6);
+draw(A--C,black+0.6);
+dot("$C$",C,2*dir(C-R)); dot("$A$",A,1.5*dir(A-L)); dot("$B$",B,dir(B-R));
+</asy>
+Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled <math>B</math> in the figure?
+<math>\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.</math>
 [[2024 AMC 12A Problems/Problem 18|Solution]]

2024 AMC 12A (Problems • Answer Key • Resources)
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Difference between revisions of "2024 AMC 12A Problems"

Revision as of 17:12, 8 November 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also