Difference between revisions of "2024 AMC 12A Problems"
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==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> | + | 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>? |
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+ | <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]] | ||
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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>? | 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>? | ||
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<math> | <math> | ||
\textbf{(A) }387 \qquad | \textbf{(A) }387 \qquad | ||
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Cyclic quadrilateral <math>ABCD</math> has lengths <math>BC=CD=3</math> and <math>DA=5</math> with <math>\angle CDA=120^\circ</math>. What is the length of the shorter diagonal of <math>ABCD</math>? | Cyclic quadrilateral <math>ABCD</math> has lengths <math>BC=CD=3</math> and <math>DA=5</math> with <math>\angle CDA=120^\circ</math>. What is the length of the shorter diagonal of <math>ABCD</math>? | ||
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<math> | <math> | ||
\textbf{(A) }\frac{31}7 \qquad | \textbf{(A) }\frac{31}7 \qquad | ||
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==Problem 20== | ==Problem 20== | ||
− | Points <math>P</math> and <math>Q</math> are chosen uniformly and independently at random on sides <math>\overline {AB}</math> and <math>\overline{AC},</math> respectively, of equilateral triangle <math>\ | + | Points <math>P</math> and <math>Q</math> are chosen uniformly and independently at random on sides <math>\overline {AB}</math> and <math>\overline{AC},</math> respectively, of equilateral triangle <math>\Delta ABC.</math> Which of the following intervals contains the probability that the area of <math>\triangle APQ</math> is less than half the area of <math>\triangle ABC?</math> |
<math>\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]</math> | <math>\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]</math> |
Revision as of 17:49, 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
Cyclic quadrilateral has lengths and with . What is the length of the shorter diagonal of ?
Problem 20
Points and are chosen uniformly and independently at random on sides and respectively, of equilateral triangle Which of the following intervals contains the probability that the area of is less than half the area of
Problem 21
Suppose that and the sequence satisfies the recurrence relation for all What is the greatest integer less than or equal to
Problem 22
The figure below shows a dotted grid cells wide and cells tall consisting of squares. Carl places -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?
Problem 23
What is the value of
Problem 24
A is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
Problem 25
A graph is about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers , where and and are not both , is the graph of symmetric about the line ?
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 |