Difference between revisions of "2024 AMC 12A Problems"
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==Problem 3== | ==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]] | [[2024 AMC 12A Problems/Problem 3|Solution]] | ||
==Problem 4== | ==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]] | [[2024 AMC 12A Problems/Problem 4|Solution]] | ||
==Problem 5== | ==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]] | [[2024 AMC 12A Problems/Problem 5|Solution]] | ||
==Problem 6== | ==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]] | [[2024 AMC 12A Problems/Problem 6|Solution]] | ||
==Problem 7== | ==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]] | [[2024 AMC 12A Problems/Problem 7|Solution]] | ||
==Problem 8== | ==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]] | [[2024 AMC 12A Problems/Problem 8|Solution]] | ||
==Problem 9== | ==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]] | [[2024 AMC 12A Problems/Problem 9|Solution]] | ||
==Problem 10== | ==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]] | [[2024 AMC 12A Problems/Problem 10|Solution]] | ||
==Problem 11== | ==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]] | [[2024 AMC 12A Problems/Problem 11|Solution]] | ||
==Problem 12== | ==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]] | [[2024 AMC 12A Problems/Problem 12|Solution]] | ||
==Problem 13== | ==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]] | [[2024 AMC 12A Problems/Problem 13|Solution]] | ||
==Problem 14== | ==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> | ||
+ | <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]] | [[2024 AMC 12A Problems/Problem 14|Solution]] | ||
==Problem 15== | ==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]] | [[2024 AMC 12A Problems/Problem 15|Solution]] | ||
==Problem 16== | ==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]] | [[2024 AMC 12A Problems/Problem 16|Solution]] | ||
==Problem 17== | ==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]] | [[2024 AMC 12A Problems/Problem 17|Solution]] | ||
==Problem 18== | ==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/Problem 18|Solution]] |
Revision as of 17:12, 8 November 2024
2024 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
[hide]- 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
What is the value of
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 where and are constants, is the time in minutes, is the length of the trail in miles, and is the altitude gain in feet. The model estimates that it will take minutes to hike to the top if a trail is miles long and ascends feet, as well as if a trail is miles long and ascends feet. How many minutes does the model estimates it will take to hike to the top if the trail is miles long and ascends feet?
Problem 3
The number 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?
Problem 4
What is the least value of such that is a multiple of ?
Problem 5
A data set containing numbers, some of which are , has mean . When all the 6s are removed, the data set has mean . How many 6s were in the original data set?
Problem 6
The product of three integers is . What is the least possible positive sum of the three integers?
Problem 7
In , and . Points lie on hypotenuse so that . What is the length of the vector sum
Problem 8
How many angles with satisfy ?
Problem 9
Let be the greatest integer such that both and are perfect squares. What is the units digit of ?
Problem 10
Let be the radian measure of the smallest angle in a right triangle. Let be the radian measure of the smallest angle in a right triangle. In terms of , what is ?
Problem 11
There are exactly positive integers with such that the base- integer is divisible by (where is in base ten). What is the sum of the digits of ?
Problem 12
The first three terms of a geometric sequence are the integers and where What is the sum of the digits of the least possible value of
Problem 13
The graph of has an axis of symmetry. What is the reflection of the point over this axis?
Problem 14
The numbers, in order, of each row and the numbers, in order, of each column of a array of integers form an arithmetic progression of length The numbers in positions and are and respectively. What number is in position
Problem 15
The roots of are and What is the value of
Problem 16
A set of tokens ---- red, white, blue, and black ---- is to be distributed at random to game players, 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 , where and are relatively prime positive integers. What is ?
Problem 17
Integers , , and satisfy , , and . What is ?
Problem 18
On top of a rectangular card with sides of length and , an identical card is placed so that two of their diagonals line up, as shown (, in this case).
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 in the figure?
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
See also
2024 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2023 AMC 12B Problems |
Followed by 2024 AMC 12B 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 12 Problems and Solutions |