Difference between revisions of "2021 AMC 12B Problems"
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+ | I know that I want about <math>\frac{2}{3}</math> of the box of integer coordinates above my line. There are a total of 30 integer coordinates in the desired range for each axis which gives a total of 900 lattice points. I estimate that the slope, m, is <math>\frac{2}{3}</math>. Now, although there is probably an easier solution, I would try to count the number of points above the line to see if there are 600 points above the line. The line <math>y=\frac{2}{3}x</math> separates the area inside the box so that <math>\frac{2}{3}</math> of the are is above the line. | ||
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+ | I find that the number of coordinates with <math>x=1</math> above the line is 30, and the number of coordinates with <math>x=2</math> above the line is 29. Every time the line <math>y=\frac{2}{3}x</math> hits a y-value with an integer coordinate, the number of points above the line decreases by one. I wrote out the sum of 30 terms in hopes of finding a pattern. I graphed the first couple positive integer x-coordinates, and found that the sum of the integers above the line is <math>30+29+28+28+27+26+26 \ldots+ 10</math>. The even integer repeats itself every third term in the sum. I found that the average of each of the terms is 20, and there are 30 of them which means that exactly 600 above the line as desired. This give a lower bound because if the slope decreases a little bit, then the points that the line goes through will be above the line. | ||
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+ | To find the upper bound, notice that each point with an integer-valued x-coordinate is either <math>\frac{1}{3}</math> or <math>\frac{2}{3}</math> above the line. Since the slope through a point is the y-coordinate divided by the x-coordinate, a shift in the slope will increase the y-value of the higher x-coordinates. We turn our attention to <math>x=28, 29, 30</math> which the line <math>y=\frac{2}{3}x</math> intersects at <math>y= \frac{56}{3}, \frac{58}{3}, 20</math>. The point (30,20) is already counted below the line, and we can clearly see that if we slowly increase the slope of the line, we will hit the point (28,19) since (28, <math>\frac{56}{3}</math>) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is <math>y=\frac{19}{28}x</math>. This gives an upper bound of <math>m=\frac{19}{28}</math>. | ||
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+ | Taking the upper bound of m and subtracting the lower bound yields <math>\frac{19}{28}-\frac{2}{3}=\frac{1}{84}</math>. This is answer E. | ||
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+ | ~Theajl | ||
==See also== | ==See also== |
Revision as of 00:26, 12 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
The figure is constructed from line segments, each of which has length
. The area of pentagon
can be written is
, where
and
are positive integers. What is
Problem 16
Let be a polynomial with leading coefficient
whose three roots are the reciprocals of the three roots of
where
What is
in terms of
and
Problem 17
Let be an isoceles trapezoid having parallel bases
and
with
Line segments from a point inside
to the vertices divide the trapezoid into four triangles whose areas are
and
starting with the triangle with base
and moving clockwise as shown in the diagram below. What is the ratio
Problem 18
Let be a complex number satisfying
What is the value of
Problem 19
Two fair dice, each with at least faces are rolled. On each face of each dice is printed a distinct integer from
to the number of faces on that die, inclusive. The probability of rolling a sum if
is
of the probability of rolling a sum of
and the probability of rolling a sum of
is
. What is the least possible number of faces on the two dice combined?
Problem 20
Let and
be the unique polynomials such that
and the degree of
is less than
What is
Problem 21
Let be the sum of all positive real numbers
for which
Which of the following statements is true?
Problem 22
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes and
can be changed into any of the following by one move:
or
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
Problem 23
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin is
for
More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is
where
and
are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins
and
) What is
Problem 24
Let be a parallelogram with area
. Points
and
are the projections of
and
respectively, onto the line
and points
and
are the projections of
and
respectively, onto the line
See the figure, which also shows the relative locations of these points.
Suppose and
and let
denote the length of
the longer diagonal of
Then
can be written in the form
where
and
are positive integers and
is not divisible by the square of any prime. What is
Problem 25
Let be the set of lattice points in the coordinate plane, both of whose coordinates are integers between
and
inclusive. Exactly
points in
lie on or below a line with equation
The possible values of
lie in an interval of length
where
and
are relatively prime positive integers. What is
I know that I want about of the box of integer coordinates above my line. There are a total of 30 integer coordinates in the desired range for each axis which gives a total of 900 lattice points. I estimate that the slope, m, is
. Now, although there is probably an easier solution, I would try to count the number of points above the line to see if there are 600 points above the line. The line
separates the area inside the box so that
of the are is above the line.
I find that the number of coordinates with above the line is 30, and the number of coordinates with
above the line is 29. Every time the line
hits a y-value with an integer coordinate, the number of points above the line decreases by one. I wrote out the sum of 30 terms in hopes of finding a pattern. I graphed the first couple positive integer x-coordinates, and found that the sum of the integers above the line is
. The even integer repeats itself every third term in the sum. I found that the average of each of the terms is 20, and there are 30 of them which means that exactly 600 above the line as desired. This give a lower bound because if the slope decreases a little bit, then the points that the line goes through will be above the line.
To find the upper bound, notice that each point with an integer-valued x-coordinate is either or
above the line. Since the slope through a point is the y-coordinate divided by the x-coordinate, a shift in the slope will increase the y-value of the higher x-coordinates. We turn our attention to
which the line
intersects at
. The point (30,20) is already counted below the line, and we can clearly see that if we slowly increase the slope of the line, we will hit the point (28,19) since (28,
) is closer to the lattice point. The slope of the line which goes through both the origin and (28,19) is
. This gives an upper bound of
.
Taking the upper bound of m and subtracting the lower bound yields . This is answer E.
~Theajl
See also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12A Problems |
Followed by 2022 AMC 12A Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.