Difference between revisions of "2021 AMC 12B Problems"
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==Problem 2== | ==Problem 2== | ||
− | + | At a math contest, <math>57</math> students are wearing blue shirts, and another <math>75</math> students are wearing yellow shirts. The <math>132</math> students are assigned into <math>66</math> pairs. In exactly <math>23</math> of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | |
+ | |||
+ | <math>\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64</math> | ||
[[2021 AMC 12B Problems/Problem 2|Solution]] | [[2021 AMC 12B Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | + | Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math> | |
+ | |||
+ | <math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math> | ||
[[2021 AMC 12B Problems/Problem 3|Solution]] | [[2021 AMC 12B Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <math>84</math>, and the afternoon class's mean score is <math>70</math>. The ratio of the number of students in the morning class to the number of students in the afternoon class is <math>\frac34</math>. What is the mean of the score of all the students? | |
+ | |||
+ | <math>\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78</math> | ||
[[2021 AMC 12B Problems/Problem 4|Solution]] | [[2021 AMC 12B Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y=-x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b-a?</math> | |
+ | |||
+ | <math>\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9</math> | ||
[[2021 AMC 12B Problems/Problem 5|Solution]] | [[2021 AMC 12B Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | An inverted cone with base radius <math>12 \text{cm}</math> and height <math>18\text{cm}</math> is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of <math>24\text{cm}</math>. What is the height in centimeters of the water in the cylinder? | |
+ | |||
+ | <math>\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6</math> | ||
[[2021 AMC 12B Problems/Problem 6|Solution]] | [[2021 AMC 12B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | Let <math>N=34\cdot34\cdot63\cdot270.</math> What is the ratio of the sum of the odd divisors of <math>N</math> to the sum of the even divisors of <math>N?</math> | |
+ | |||
+ | <math>\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3</math> | ||
[[2021 AMC 12B Problems/Problem 7|Solution]] | [[2021 AMC 12B Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | Three equally spaced parallel lines intersect a circle, creating three chords of lengths <math>38,38,</math> and <math>34</math>. What is the distance between two adjacent parallel lines? | |
+ | |||
+ | <math>\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12</math> | ||
[[2021 AMC 12B Problems/Problem 8|Solution]] | [[2021 AMC 12B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | + | What is the value of<cmath>\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?</cmath> | |
+ | <math>\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }\frac54 \qquad \textbf{(D) }2 \qquad \textbf{(E) }\log_2 5</math> | ||
[[2021 AMC 12B Problems/Problem 9|Solution]] | [[2021 AMC 12B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | + | Two distinct numbers are selected from the set <math>\{1,2,3,4,\dots,36,37\}</math> so that the sum of the remaining <math>35</math> numbers is the product of these two numbers. What is the difference of these two numbers? | |
+ | |||
+ | <math>\textbf{(A) }5 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8\qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math> | ||
[[2021 AMC 12B Problems/Problem 10|Solution]] | [[2021 AMC 12B Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | + | Triangle <math>ABC</math> has <math>AB=13,BC=14</math> and <math>AC=15</math>. Let <math>P</math> be the point on <math>\overline{AC}</math> such that <math>PC=10</math>. There are exactly two points <math>D</math> and <math>E</math> on line <math>BP</math> such that quadrilaterals <math>ABCD</math> and <math>ABCE</math> are trapezoids. What is the distance <math>DE?</math> | |
+ | |||
+ | <math>\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18</math> | ||
[[2021 AMC 12B Problems/Problem 11|Solution]] | [[2021 AMC 12B Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the greatest integer is then returned to the set, the average value of the integers rises to <math>40</math>. The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S</math>? | |
+ | |||
+ | <math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math> | ||
[[2021 AMC 12B Problems/Problem 12|Solution]] | [[2021 AMC 12B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | + | How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath> | |
+ | <math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math> | ||
[[2021 AMC 12B Problems/Problem 13|Solution]] | [[2021 AMC 12B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | Let <math>ABCD</math> be a rectangle and let <math>\overline{DM}</math> be a segment perpendicular to the plane of <math>ABCD</math>. Suppose that <math>\overline{DM}</math> has integer length, and the lengths of <math>\overline{MA},\overline{MC},</math> and <math>\overline{MB}</math> are consecutive odd positive integers (in this order). What is the volume of pyramid <math>MABCD?</math> | |
+ | |||
+ | <math>\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}</math> | ||
[[2021 AMC 12B Problems/Problem 14|Solution]] | [[2021 AMC 12B Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | + | The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written as <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math> | |
+ | <asy> /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,dir(0)); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); </asy> | ||
+ | <math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math> | ||
[[2021 AMC 12B Problems/Problem 15|Solution]] | [[2021 AMC 12B Problems/Problem 15|Solution]] | ||
==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 is <math>g(1)</math> in terms of <math>a,b,</math> and <math>c?</math> | |
+ | |||
+ | <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]] | ||
==Problem 17== | ==Problem 17== | ||
− | + | Let <math>ABCD</math> be an isosceles trapezoid having parallel bases <math>\overline{AB}</math> and <math>\overline{CD}</math> with <math>AB>CD.</math> Line segments from a point inside <math>ABCD</math> to the vertices divide the trapezoid into four triangles whose areas are <math>2, 3, 4,</math> and <math>5</math> starting with the triangle with base <math>\overline{CD}</math> and moving clockwise as shown in the diagram below. What is the ratio <math>\frac{AB}{CD}?</math> | |
+ | <asy> | ||
+ | unitsize(100); | ||
+ | pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); | ||
+ | draw(A--B--C--D--cycle, black); | ||
+ | draw(A--P, black); | ||
+ | draw(B--P, black); | ||
+ | draw(C--P, black); | ||
+ | draw(D--P, black); | ||
+ | label("$A$",A,(-1,0)); | ||
+ | label("$B$",B,(1,0)); | ||
+ | label("$C$",C,(1,-0)); | ||
+ | label("$D$",D,(-1,0)); | ||
+ | label("$2$",E,(0,0)); | ||
+ | label("$3$",F,(0,0)); | ||
+ | label("$4$",G,(0,0)); | ||
+ | label("$5$",H,(0,0)); | ||
+ | dot(A^^B^^C^^D^^P); | ||
+ | </asy> | ||
+ | <math>\textbf{(A)}\: 3\qquad\textbf{(B)}\: 2+\sqrt{2}\qquad\textbf{(C)}\: 1+\sqrt{6}\qquad\textbf{(D)}\: 2\sqrt{3}\qquad\textbf{(E)}\: 3\sqrt{2}</math> | ||
[[2021 AMC 12B Problems/Problem 17|Solution]] | [[2021 AMC 12B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | + | Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value of <math>z+\frac 6z?</math> | |
+ | |||
+ | <math>\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4</math> | ||
[[2021 AMC 12B Problems/Problem 18|Solution]] | [[2021 AMC 12B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | + | Two fair dice, each with at least <math>6</math> faces are rolled. On each face of each die is printed a distinct integer from <math>1</math> to the number of faces on that die, inclusive. The probability of rolling a sum of <math>7</math> is <math>\frac34</math> of the probability of rolling a sum of <math>10,</math> and the probability of rolling a sum of <math>12</math> is <math>\frac{1}{12}</math>. What is the least possible number of faces on the two dice combined? | |
+ | |||
+ | <math>\textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18\qquad \textbf{(D) }19 \qquad \textbf{(E) }20</math> | ||
[[2021 AMC 12B Problems/Problem 19|Solution]] | [[2021 AMC 12B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | + | Let <math>Q(z)</math> and <math>R(z)</math> be the unique polynomials such that<cmath>z^{2021}+1=(z^2+z+1)Q(z)+R(z)</cmath>and the degree of <math>R</math> is less than <math>2.</math> What is <math>R(z)?</math> | |
+ | |||
+ | <math>\textbf{(A) }{-}z \qquad \textbf{(B) }{-}1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1</math> | ||
[[2021 AMC 12B Problems/Problem 20|Solution]] | [[2021 AMC 12B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | + | Let <math>S</math> be the sum of all positive real numbers <math>x</math> for which<cmath>x^{2^{\sqrt2}}=\sqrt2^{2^x}.</cmath>Which of the following statements is true? | |
+ | |||
+ | <math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math> | ||
[[2021 AMC 12B Problems/Problem 21|Solution]] | [[2021 AMC 12B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==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 <math>4</math> and <math>2</math> can be changed into any of the following by one move: <math>(3,2),(2,1,2),(4),(4,1),(2,2),</math> or <math>(1,1,2).</math> | |
+ | |||
+ | <asy> unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(",",(20,0)); label(",",(29,0)); label(",...",(35.5,0)); </asy> | ||
+ | |||
+ | 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? | ||
+ | |||
+ | <math>\textbf{(A) }(6,1,1) \qquad \textbf{(B) }(6,2,1) \qquad \textbf{(C) }(6,2,2)\qquad \textbf{(D) }(6,3,1) \qquad \textbf{(E) }(6,3,2)</math> | ||
[[2021 AMC 12B Problems/Problem 22|Solution]] | [[2021 AMC 12B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | + | Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin <math>i</math> is <math>2^{-i}</math> for <math>i=1,2,3,....</math> More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is <math>\frac pq,</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins <math>3,17,</math> and <math>10.</math>) What is <math>p+q?</math> | |
+ | |||
+ | <math>\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59</math> | ||
[[2021 AMC 12B Problems/Problem 23|Solution]] | [[2021 AMC 12B Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
− | + | Let <math>ABCD</math> be a parallelogram with area <math>15</math>. Points <math>P</math> and <math>Q</math> are the projections of <math>A</math> and <math>C,</math> respectively, onto the line <math>BD;</math> and points <math>R</math> and <math>S</math> are the projections of <math>B</math> and <math>D,</math> respectively, onto the line <math>AC.</math> See the figure, which also shows the relative locations of these points. | |
+ | |||
+ | <asy> | ||
+ | size(350); | ||
+ | defaultpen(linewidth(0.8)+fontsize(11)); | ||
+ | real theta = aTan(1.25/2); | ||
+ | pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; | ||
+ | draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); | ||
+ | draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); | ||
+ | dot("$A$",A,dir(270)); | ||
+ | dot("$B$",B,E); | ||
+ | dot("$C$",C,N); | ||
+ | dot("$D$",D,W); | ||
+ | dot("$P$",P,SE); | ||
+ | dot("$Q$",Q,NE); | ||
+ | dot("$R$",R,N); | ||
+ | dot("$S$",S,dir(270)); | ||
+ | </asy> | ||
+ | |||
+ | Suppose <math>PQ=6</math> and <math>RS=8,</math> and let <math>d</math> denote the length of <math>\overline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. What is <math>m+n+p?</math> | ||
+ | |||
+ | <math>\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113</math> | ||
[[2021 AMC 12B Problems/Problem 24|Solution]] | [[2021 AMC 12B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
− | + | Let <math>S</math> be the set of lattice points in the coordinate plane, both of whose coordinates are integers between <math>1</math> and <math>30,</math> inclusive. Exactly <math>300</math> points in <math>S</math> lie on or below a line with equation <math>y=mx.</math> The possible values of <math>m</math> lie in an interval of length <math>\frac ab,</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a+b?</math> | |
+ | |||
+ | <math>\textbf{(A) }31 \qquad \textbf{(B) }47 \qquad \textbf{(C) }62\qquad \textbf{(D) }72 \qquad \textbf{(E) }85</math> | ||
[[2021 AMC 12B Problems/Problem 25|Solution]] | [[2021 AMC 12B Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
− | {{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[ | + | {{AMC12 box|year=2021|ab=B|before=[[2021 AMC 12A Problems]]|after=[[2021 Fall AMC 12A Problems]]}} |
[[Category:AMC 12 Problems]] | [[Category:AMC 12 Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:31, 27 October 2023
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
pairs. In exactly
of these pairs, both students are wearing blue shirts. In how many pairs are both students 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 class 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 greatest 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 as
, 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 isosceles 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 die is printed a distinct integer from
to the number of faces on that die, inclusive. The probability of rolling a sum of
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 independently 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
See also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2021 AMC 12A Problems |
Followed by 2021 Fall AMC 12A Problems |
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All AMC 12 Problems and Solutions |
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