Difference between revisions of "2021 Fall AMC 12B Problems"

Latest revision as of 00:07, 14 November 2024

2021 Fall AMC 12B (Answer Key) Printable versions: Wiki • Fall AoPS Resources • Fall PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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

Problem 1

What is the value of $1234+2341+3412+4123?$

$\textbf{(A)}\: 10{,}000\qquad\textbf{(B)} \: 10{,}010\qquad\textbf{(C)} \: 10{,}110\qquad\textbf{(D)} \: 11{,}000\qquad\textbf{(E)} \: 11{,}110$

Solution

Problem 2

What is the area of the shaded figure shown below? $[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]$

$\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12$

Solution

Problem 3

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$

$\textbf{(A)}\: 10\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120$

Solution

Problem 4

Let $n=8^{2022}$ . Which of the following is equal to $\frac{n}{4}?$

$\textbf{(A)}\: 4^{1010}\qquad\textbf{(B)} \: 2^{2022}\qquad\textbf{(C)} \: 8^{2018}\qquad\textbf{(D)} \: 4^{3031}\qquad\textbf{(E)} \: 4^{3032}$

Solution

Problem 5

Call a fraction $\frac{a}{b}$ , not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$ . How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution

Problem 6

The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$ . What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$ ?

$\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22$

Solution

Problem 7

Which of the following conditions is sufficient to guarantee that integers $x$ , $y$ , and $z$ satisfy the equation $x(x-y)+y(y-z)+z(z-x) = 1?$

$\textbf{(A)} \: x>y$ and $y=z$

$\textbf{(B)} \: x=y-1$ and $y=z-1$

$\textbf{(C)} \: x=z+1$ and $y=x+1$

$\textbf{(D)} \: x=z$ and $y-1=x$

$\textbf{(E)} \: x+y+z=1$

Solution

Problem 8

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?

$\textbf{(A)} \: 105 \qquad\textbf{(B)} \: 120 \qquad\textbf{(C)} \: 135 \qquad\textbf{(D)} \: 150 \qquad\textbf{(E)} \: 165$

Solution

Problem 9

Triangle $ABC$ is equilateral with side length $6$ . Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$ , $O$ , and $C$ ?

$\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi$

Solution

Problem 10

What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)$ is isosceles?

$\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380$

Solution

Problem 11

Una rolls $6$ standard $6$ -sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$

$\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Solution

Problem 12

For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\div 14=\frac{12}{7}$ What is $f(768)-f(384)?$

$\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}$

Solution

Problem 13

Let $c = \frac{2\pi}{11}.$ What is the value of $\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$

$\textbf{(A)}\ {-}1 \qquad\textbf{(B)}\ {-}\frac{\sqrt{11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\ \frac{10}{11} \qquad\textbf{(E)}\ 1$

Solution

Problem 14

Suppose that $P(z), Q(z)$ , and $R(z)$ are polynomials with real coefficients, having degrees $2$ , $3$ , and $6$ , respectively, and constant terms $1$ , $2$ , and $3$ , respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$ . What is the minimum possible value of $N$ ?

$\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

Solution

Problem 15

Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$ -sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$ ?

$[asy] defaultpen(fontsize(8)+0.8); size(150); pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; real x=45, y=90, z=60; O=origin; A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); draw(A1--A2--A3--A4--A1, gray+0.25+dashed); filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); dot(O); pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4); P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1); P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2); P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3); R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4); R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); draw(A1--P1--B2--R1--C3--Q1--A2); draw(A2--P2--B3--R2--C4--Q2--A3); draw(A3--P3--B4--R3--C1--Q3--A4); draw(A4--P4--B1--R4--C2--Q4--A1); [/asy]$

$(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147$

Solution

Problem 16

Suppose $a$ , $b$ , $c$ are positive integers such that $a+b+c=23$ and $\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.$ What is the sum of all possible distinct values of $a^2+b^2+c^2$ ?

$\textbf{(A)}\: 259\qquad\textbf{(B)} \: 438\qquad\textbf{(C)} \: 516\qquad\textbf{(D)} \: 625\qquad\textbf{(E)} \: 687$

Solution

Problem 17

A bug starts at a vertex of a grid made of equilateral triangles of side length $1$ . At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?

$\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\ \frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}$

Solution

Problem 18

Set $u_0 = \frac{1}{4}$ , and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$

This sequence tends to a limit; call it $L$ . What is the least value of $k$ such that $|u_k-L| \le \frac{1}{2^{1000}}?$

$\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401$

Solution

Problem 19

Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

$(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68$

Solution

Problem 20

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

$(\textbf{A})\: 7\qquad(\textbf{B}) \: 8\qquad(\textbf{C}) \: 9\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 11$

Solution

Problem 21

For real numbers $x$ , let $P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$ where $i = \sqrt{-1}$ . For how many values of $x$ with $0\leq x<2\pi$ does $P(x)=0?$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

Problem 22

Right triangle $ABC$ has side lengths $BC=6$ , $AC=8$ , and $AB=10$ . A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$ . A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$ . What is $OP$ ?

$\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ \frac{73}{25} \qquad\textbf{(E)}\ 3$

Solution

Problem 23

What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$ ? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)

$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{29}{30} \qquad\textbf{(E)}\ 1$

Solution

Problem 24

Triangle $ABC$ has side lengths $AB = 11, BC=24$ , and $CA = 20$ . The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$ , and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$ . The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$ . What is $CF$ ?

$\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}$

Solution

Problem 25

For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ , and $10$ . For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$ . How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution

@@ Line 4: / Line 4: @@
 What is the value of <math>1234+2341+3412+4123?</math>
-<math>(\textbf{A})\: 10{,}000\qquad(\textbf{B}) \: 10{,}010\qquad(\textbf{C}) \: 10{,}110\qquad(\textbf{D}) \: 11{,}000\qquad(\textbf{E}) \: 11{,}110</math>
+<math>\textbf{(A)}\: 10{,}000\qquad\textbf{(B)} \: 10{,}010\qquad\textbf{(C)} \: 10{,}110\qquad\textbf{(D)} \: 11{,}000\qquad\textbf{(E)} \: 11{,}110</math>
 [[2021 Fall AMC 12B Problems/Problem 1|Solution]]
@@ Line 31: / Line 31: @@
 }
 label("$0$", O, 2*SW);
-draw(O--X+(0.15,0), EndArrow);
+draw(O--X+(0.35,0), black+1.5, EndArrow(10));
-draw(O--Y+(0,0.15), EndArrow);
+draw(O--Y+(0,0.35), black+1.5, EndArrow(10));
 draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
 </asy>
-<math>(\textbf{A})\: 4\qquad(\textbf{B}) \: 6\qquad(\textbf{C}) \: 8\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 12</math>
+<math>\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12</math>
 [[2021 Fall AMC 12B Problems/Problem 2|Solution]]
@@ Line 43: / Line 43: @@
 At noon on a certain day, Minneapolis is <math>N</math> degrees warmer than St. Louis. At <math>4{:}00</math> the temperature in Minneapolis has fallen by <math>5</math> degrees while the temperature in St. Louis has risen by <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <math>N?</math>
-<math>(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120</math>
+<math>\textbf{(A)}\: 10\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120</math>
 [[2021 Fall AMC 12B Problems/Problem 3|Solution]]
@@ Line 50: / Line 50: @@
 Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math>
-<math>(\textbf{A})\: 4^{1010}\qquad(\textbf{B}) \: 2^{2022}\qquad(\textbf{C}) \: 8^{2018}\qquad(\textbf{D}) \: 4^{3031}\qquad(\textbf{E}) \: 4^{3032}</math>
+<math>\textbf{(A)}\: 4^{1010}\qquad\textbf{(B)} \: 2^{2022}\qquad\textbf{(C)} \: 8^{2018}\qquad\textbf{(D)} \: 4^{3031}\qquad\textbf{(E)} \: 4^{3032}</math>
 [[2021 Fall AMC 12B Problems/Problem 4|Solution]]
@@ Line 62: / Line 62: @@
 ==Problem 6==
-The largest prime factor of <math>16384</math> is <math>2</math> because <math>16384 = 2^{14}</math>. What is the sum of the digits of  the greatest prime number that is a divisor of <math>16383</math>?
+The greatest prime number that is a divisor of <math>16{,}384</math> is <math>2</math> because <math>16{,}384 = 2^{14}</math>. What is the sum of the digits of the greatest prime number that is a divisor of <math>16{,}383</math>?
 <math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math>
-[[2021 Fall AMC 12B Problems/Problem 7|Solution]]
+[[2021 Fall AMC 12B Problems/Problem 6|Solution]]
 ==Problem 7==
@@ Line 73: / Line 73: @@
 <math>\textbf{(A)} \: x>y</math> and <math>y=z</math>
 <math>\textbf{(B)} \: x=y-1</math> and <math>y=z-1</math>
 <math>\textbf{(C)} \: x=z+1</math> and <math>y=x+1</math>
 <math>\textbf{(D)} \: x=z</math> and <math>y-1=x</math>
 <math>\textbf{(E)} \: x+y+z=1</math>
@@ Line 89: / Line 93: @@
 ==Problem 9==
+Triangle <math>ABC</math> is equilateral with side length <math>6</math>. Suppose that <math>O</math> is the center of the inscribed
+circle of this triangle. What is the area of the circle passing through <math>A</math>, <math>O</math>, and <math>C</math>?
+<math>\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi</math>
 [[2021 Fall AMC 12B Problems/Problem 9|Solution]]
@@ Line 94: / Line 102: @@
 ==Problem 10==
+What is the sum of all possible values of <math>t</math> between <math>0</math> and <math>360</math> such that the triangle in the coordinate plane whose vertices are <cmath>(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)</cmath>
+is isosceles?
+<math>\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380</math>
 [[2021 Fall AMC 12B Problems/Problem 10|Solution]]
@@ Line 105: / Line 117: @@
 ==Problem 12==
+For <math>n</math> a positive integer, let <math>f(n)</math> be the quotient obtained when the sum of all positive divisors of <math>n</math> is divided by <math>n.</math> For example, <cmath>f(14)=(1+2+7+14)\div 14=\frac{12}{7}</cmath>
+What is <math>f(768)-f(384)?</math>
+<math>\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\
+\frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}</math>
 [[2021 Fall AMC 12B Problems/Problem 12|Solution]]
 ==Problem 13==
+Let <math>c = \frac{2\pi}{11}.</math> What is the value of
+<cmath>\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?</cmath>
+<math>\textbf{(A)}\ {-}1 \qquad\textbf{(B)}\ {-}\frac{\sqrt{11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\
+\frac{10}{11} \qquad\textbf{(E)}\ 1</math>
 [[2021 Fall AMC 12B Problems/Problem 13|Solution]]
 ==Problem 14==
+Suppose that <math>P(z), Q(z)</math>, and <math>R(z)</math> are polynomials with real coefficients, having degrees <math>2</math>, <math>3</math>, and <math>6</math>, respectively, and constant terms <math>1</math>, <math>2</math>, and <math>3</math>, respectively. Let <math>N</math> be the number of distinct complex numbers <math>z</math> that satisfy the equation <math>P(z) \cdot Q(z)=R(z)</math>. What is the minimum possible value of <math>N</math>?
+<math>\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5</math>
 [[2021 Fall AMC 12B Problems/Problem 14|Solution]]
@@ Line 121: / Line 143: @@
 ==Problem 15==
 Three identical square sheets of paper each with side length <math>6</math> are stacked on top of each other. The middle sheet is rotated clockwise <math>30^\circ</math> about its center and the top sheet is rotated clockwise <math>60^\circ</math> about its center, resulting in the <math>24</math>-sided polygon shown in the figure below. The area of this polygon can be expressed in the form <math>a-b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. What is <math>a+b+c</math>?
+<center><asy>
-IMAGE
+defaultpen(fontsize(8)+0.8); size(150);
+pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4;
+real x=45, y=90, z=60; O=origin;
+A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y);
+B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z);
+C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z);
+draw(A1--A2--A3--A4--A1, gray+0.25+dashed);
+filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25));
+filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25));
+dot(O);
+pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4;
+P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4);
+P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1);
+P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2);
+P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3);
+R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4);
+R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2);
+draw(A1--P1--B2--R1--C3--Q1--A2);
+draw(A2--P2--B3--R2--C4--Q2--A3);
+draw(A3--P3--B4--R3--C1--Q3--A4);
+draw(A4--P4--B1--R4--C2--Q4--A1);
+</asy></center>
 <math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math>
@@ Line 130: / Line 172: @@
 ==Problem 16==
+Suppose <math>a</math>, <math>b</math>, <math>c</math> are positive integers such that <cmath>a+b+c=23</cmath> and <cmath>\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.</cmath> What is the sum of all possible distinct values of <math>a^2+b^2+c^2</math>?
+<math>\textbf{(A)}\: 259\qquad\textbf{(B)} \: 438\qquad\textbf{(C)} \: 516\qquad\textbf{(D)} \: 625\qquad\textbf{(E)} \: 687</math>
 [[2021 Fall AMC 12B Problems/Problem 16|Solution]]
@@ Line 143: / Line 188: @@
 ==Problem 18==
+Set <math>u_0 = \frac{1}{4}</math>, and for <math>k \ge 0</math> let <math>u_{k+1}</math> be determined by the recurrence <cmath>u_{k+1} = 2u_k - 2u_k^2.</cmath>
+This sequence tends to a limit; call it <math>L</math>. What is the least value of <math>k</math> such that <cmath>|u_k-L| \le \frac{1}{2^{1000}}?</cmath>
+<math>\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401</math>
 [[2021 Fall AMC 12B Problems/Problem 18|Solution]]
 ==Problem 19==
-Regular polygons with <math>5</math>, <math>6</math>, <math>7</math>, and <math>8</math> sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
+Regular polygons with <math>5,6,7,</math> and <math>8</math> sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
 <math>(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68</math>
@@ Line 162: / Line 212: @@
 ==Problem 21==
+For real numbers <math>x</math>, let
+<cmath>P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)</cmath>
+where <math>i = \sqrt{-1}</math>. For how many values of <math>x</math> with <math>0\leq x<2\pi</math> does
+<cmath>P(x)=0?</cmath>
+<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\  1 \qquad\textbf{(C)}\  2 \qquad\textbf{(D)}\
+\qquad\textbf{(E)}\ 4</math>
 [[2021 Fall AMC 12B Problems/Problem 21|Solution]]
@@ Line 167: / Line 224: @@
 ==Problem 22==
+Right triangle <math>ABC</math> has side lengths <math>BC=6</math>, <math>AC=8</math>, and <math>AB=10</math>. A circle centered at <math>O</math> is tangent to line <math>BC</math> at <math>B</math> and passes through <math>A</math>. A circle centered at <math>P</math> is tangent to line <math>AC</math> at <math>A</math> and passes through <math>B</math>. What is <math>OP</math>?
+<math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\  \frac{29}{10} \qquad\textbf{(C)}\  \frac{35}{12} \qquad\textbf{(D)}\
+\frac{73}{25} \qquad\textbf{(E)}\ 3</math>
 [[2021 Fall AMC 12B Problems/Problem 22|Solution]]
 ==Problem 23==
+What is the average number of pairs of consecutive integers in a randomly selected subset of <math>5</math> distinct integers chosen from the set <math>\{ 1, 2, 3, …, 30\}</math>? (For example the set <math>\{1, 17, 18, 19, 30\}</math> has <math>2</math> pairs of consecutive integers.)
+<math>\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
+\frac{29}{30} \qquad\textbf{(E)}\ 1</math>
 [[2021 Fall AMC 12B Problems/Problem 23|Solution]]
@@ Line 177: / Line 241: @@
 ==Problem 24==
+Triangle <math>ABC</math> has side lengths <math>AB = 11, BC=24</math>, and <math>CA = 20</math>. The bisector of <math>\angle{BAC}</math> intersects <math>\overline{BC}</math> in point <math>D</math>, and intersects the circumcircle of <math>\triangle{ABC}</math> in point <math>E \ne A</math>. The circumcircle of <math>\triangle{BED}</math> intersects the line <math>AB</math> in points <math>B</math> and <math>F \ne B</math>. What is <math>CF</math>?
+<math>\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}</math>
 [[2021 Fall AMC 12B Problems/Problem 24|Solution]]
@@ Line 182: / Line 249: @@
 ==Problem 25==
+For <math>n</math> a positive integer, let <math>R(n)</math> be the sum of the remainders when <math>n</math> is divided by <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, <math>9</math>, and <math>10</math>. For example, <math>R(15) = 1+0+3+0+3+1+7+6+5=26</math>. How many two-digit positive integers <math>n</math> satisfy <math>R(n) = R(n+1)\,?</math>
+<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4</math>
 [[2021 Fall AMC 12B Problems/Problem 25|Solution]]
 ==See also==
-{{AMC12 box|year=2021 Fall|ab=A|before=[[2021 Fall AMC 12B Problems]]}}
+{{AMC12 box|year=2021 Fall|ab=B|before=[[2021 Fall AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}
 {{MAA Notice}}

2021 Fall AMC 12B (Problems • Answer Key • Resources)
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Difference between revisions of "2021 Fall AMC 12B Problems"

Latest revision as of 00:07, 14 November 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also