Difference between revisions of "2021 AMC 12B Problems"
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==Problem 16== | ==Problem 16== | ||
− | + | Let <math>g(x)</math> be a polynomial with leading coefficient <math>1,</math> whose three roots are the reciprocals of the three roots of <math>f(x)=x^3+ax^2+bx+c,</math> where <math>1<a<b<c.</math> What iis <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math> | |
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+ | <math>\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}</math> | ||
[[2021 AMC 12B Problems/Problem 16|Solution]] | [[2021 AMC 12B Problems/Problem 16|Solution]] |
Revision as of 15:49, 11 February 2021
2021 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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
How many integer values of satisfy
Problem 2
At a math contest, students are wearing blue shirts, and another
students are wearing yellow shirts. The
students are assigned into
points. In exactly
of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts?
Problem 3
SupposeWhat is the value of
Problem 4
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is , and the afternoon class's mean score is
. The ratio of the number of students in the morning clas to the number of students in the afternoon class is
. What is the mean of the score of all the students?
Problem 5
The point in the
-plane is first rotated counterclockwise by
around the point
and then reflected about the line
. The image of
after these two transformations is at
. What is
Problem 6
An inverted cone with base radius and height
is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of
. What is the height in centimeters of the water in the cylinder?
Problem 7
Let What is the ratio of the sum of the odd divisors of
to the sum of the even divisors of
Problem 8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths and
. What is the distance between two adjacent parallel lines?
Problem 9
What is the value of
Problem 10
Two distinct numbers are selected from the set so that the sum of the remaining
numbers is the product of these two numbers. What is the difference of these two numbers?
Problem 11
Triangle has
and
. Let
be the point on
such that
. There are exactly two points
and
on line
such that quadrilaterals
and
are trapezoids. What is the distance
Problem 12
Suppose that is a finite set of positive integers. If the greatest integer in
is removed from
, then the average value (arithmetic mean) of the integers remaining is
. If the least integer in
is also removed, then the average value of the integers remaining is
. If the great integer is then returned to the set, the average value of the integers rises to
The greatest integer in the original set
is
greater than the least integer in
. What is the average value of all the integers in the set
Problem 13
How many values of in the interval
satisfy
Problem 14
Let be a rectangle and let
be a segment perpendicular to the plane of
. Suppose that
has integer length, and the lengths of
and
are consecutive odd positive integers (in this order). What is the volume of pyramid
Problem 15
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 16
Let be a polynomial with leading coefficient
whose three roots are the reciprocals of the three roots of
where
What iis
in terms of
and
Problem 17
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 18
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 19
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 20
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 21
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 22
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 23
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 24
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
Problem 25
These problems will not be posted until the 2021 AMC 12B is released on Wednesday, February 10, 2021.
See also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12A Problems |
Followed by 2022 AMC 12A 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.