Difference between revisions of "2021 AMC 12A Problems"
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==Problem 21== | ==Problem 21== | ||
− | The five solutions to the equation<cmath>(z-1)(z^2+2z+4)(z^2+4z+6)=0</cmath>may be written in the form <math>x_k+y_ki</math> for <math>1\le k\le 5,</math> where <math>x_k</math> and <math>y_k</math> are real. Let <math>\mathcal E</math> be the unique ellipse that passes through the points <math>(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),</math> and <math>(x_5,y_5)</math>. The | + | The five solutions to the equation<cmath>(z-1)(z^2+2z+4)(z^2+4z+6)=0</cmath>may be written in the form <math>x_k+y_ki</math> for <math>1\le k\le 5,</math> where <math>x_k</math> and <math>y_k</math> are real. Let <math>\mathcal E</math> be the unique ellipse that passes through the points <math>(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),</math> and <math>(x_5,y_5)</math>. The eccentricity of <math>\mathcal E</math> can be written in the form <math>\sqrt{\frac mn}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? (Recall that the eccentricity of an ellipse <math>\mathcal E</math> is the ratio <math>\frac ca</math>, where <math>2a</math> is the length of the major axis of <math>E</math> and <math>2c</math> is the is the distance between its two foci.) |
<math>\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15</math> | <math>\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15</math> |
Revision as of 23:23, 13 February 2021
2021 AMC 12A (Answer Key) Printable version: | 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
- 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
Under what conditions does hold, where and are real numbers?
It is never true.
It is true if and only if .
It is true if and only if .
It is true if and only if and .
It is always true.
Problem 3
The sum of two natural numbers is . One of the two numbers is divisible by . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Problem 4
Tom has a collection of snakes, of which are purple and of which are happy. He observes that
all of his happy snakes can add,
none of his purple snakes can subtract, and
all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Problem 5
When a student multiplied the number by the repeating decimalwhere and are digits, he did not notice the notation and just multiplied times . Later he found that his answer is less than the correct answer. What is the -digit number
Problem 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is . When black cards are added to the deck, the probability of choosing red becomes . How many cards were in the deck originally?
Problem 7
What is the least possible value of for all real numbers and
Problem 8
A sequence of numbers is defined by and for . What are the parities (evenness or oddness) of the triple of numbers , where denotes even and denotes odd?
Problem 9
Which of the following is equilvalent to
Problem 10
Two right circular cones with vertices facing down as shown in the figure below contains the same amount of liquid. The radii of the tops of the liquid surfaces are cm and cm. Into each cone is dropped a spherical marble of radius cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
Problem 11
A laser is placed at the point . The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the -axis, then hit and bounce off the -axis, then hit the point . What is the total distance the beam will travel along this path?
Problem 12
All the roots of the polynomial are positive integers, possibly repeated. What is the value of ?
Problem 13
Of the following complex numbers , which one has the property that has the greatest real part?
Problem 14
What is the value of
Problem 15
A choir direction must select a group of singers from among his tenors and basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of , and the group must have at least one singer. Let be the number of different groups that could be selected. What is the remainder when is divided by ?
Problem 16
In the following list of numbers, the integer appears times in the list for .What is the median of the numbers in this list?
Problem 17
Trapezoid has , and . Let be the intersection of the diagonals and , and let be the midpoint of . Given that , the length of can be written in the form , where and are positive integers and is not divisible by the quare of any prime. What is ?
Problem 18
Let be a function defined on the set of positive rational numbers with the property that for all positive rational numbers and . Furthermore, suppose that also has the property that for every prime number . For which of the following numbers is ?
Problem 19
How many solutions does the equation have in the closed interval ?
Problem 20
Suppose that on a parabola with vertex and a focus there exists a point such that and . What is the sum of all possible values of the length
Problem 21
The five solutions to the equationmay be written in the form for where and are real. Let be the unique ellipse that passes through the points and . The eccentricity of can be written in the form where and are relatively prime positive integers. What is ? (Recall that the eccentricity of an ellipse is the ratio , where is the length of the major axis of and is the is the distance between its two foci.)
Problem 22
Suppose that the roots of the polynomial are and , where angles are in radians. What is ?
Problem 23
Frieda the frog begins a sequence of hops on a grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
Problem 24
Semicircle has diameter of length . Circle lies tangent to at a point and intersects at points and . If and , then the area of equals , where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. What is ?
Problem 25
Let denote the number of positive integers that divide , including and . For example, and . (This function is known as the divisor function.) LetThere is a unique positive integer such that for all positive integers . What is the sum of the digits of
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2020 AMC 12B Problems |
Followed by 2021 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.