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| − | {{AMC12 Problems|year=2021 Fall|ab=B}} | + | {{AMC12 Problems|year=2024|ab=B}} |
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| | ==Problem 1== | | ==Problem 1== |
| − | What is the value of <math>1234+2341+3412+4123?</math>
| + | [[2024 AMC 12B Problems/Problem 1|Solution]] |
| − | | |
| − | <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>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 1|Solution]] | |
| | | | |
| | ==Problem 2== | | ==Problem 2== |
| − | What is the area of the shaded figure shown below?
| + | [[2024 AMC 12B Problems/Problem 2|Solution]] |
| − | <asy>
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| − | size(200);
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| − | defaultpen(linewidth(0.4)+fontsize(12));
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| − | pen s = linewidth(0.8)+fontsize(8);
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| − | | |
| − | pair O,X,Y;
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| − | O = origin;
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| − | X = (6,0);
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| − | Y = (0,5);
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| − | fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2));
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| − | for (int i=1; i<7; ++i)
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| − | {
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| − | draw((i,0)--(i,5), gray+dashed);
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| − | label("${"+string(i)+"}$", (i,0), 2*S);
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| − | if (i<6)
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| − | {
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| − | draw((0,i)--(6,i), gray+dashed);
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| − | label("${"+string(i)+"}$", (0,i), 2*W);
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| − | }
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| − | }
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| − | label("$0$", O, 2*SW);
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| − | draw(O--X+(0.35,0), black+1.5, EndArrow(10));
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| − | draw(O--Y+(0,0.35), black+1.5, EndArrow(10));
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| − | draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
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| − | </asy>
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| − | | |
| − | <math>\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 2|Solution]] | |
| | | | |
| | ==Problem 3== | | ==Problem 3== |
| − | 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>
| + | [[2024 AMC 12B Problems/Problem 3|Solution]] |
| − | | |
| − | <math>\textbf{(A)}\: 10\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 3|Solution]] | |
| | | | |
| | ==Problem 4== | | ==Problem 4== |
| − | Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math>
| + | [[2024 AMC 12B Problems/Problem 4|Solution]] |
| − | | |
| − | <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>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 4|Solution]] | |
| | | | |
| | ==Problem 5== | | ==Problem 5== |
| − | Call a fraction <math>\frac{a}{b}</math>, not necessarily in the simplest form, ''special'' if <math>a</math> and <math>b</math> are positive integers whose sum is <math>15</math>. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
| + | [[2024 AMC 12B Problems/Problem 5|Solution]] |
| − | | |
| − | <math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 5|Solution]] | |
| | | | |
| | ==Problem 6== | | ==Problem 6== |
| − | 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>?
| + | [[2024 AMC 12B Problems/Problem 6|Solution]] |
| − | | |
| − | <math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 6|Solution]] | |
| | | | |
| | ==Problem 7== | | ==Problem 7== |
| − | Which of the following conditions is sufficient to guarantee that integers <math>x</math>, <math>y</math>, and <math>z</math> satisfy the equation
| + | [[2024 AMC 12B Problems/Problem 7|Solution]] |
| − | <cmath>x(x-y)+y(y-z)+z(z-x) = 1?</cmath>
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| − | | |
| − | <math>\textbf{(A)} \: x>y</math> and <math>y=z</math>
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| − | | |
| − | <math>\textbf{(B)} \: x=y-1</math> and <math>y=z-1</math>
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| − | | |
| − | <math>\textbf{(C)} \: x=z+1</math> and <math>y=x+1</math>
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| − | | |
| − | <math>\textbf{(D)} \: x=z</math> and <math>y-1=x</math>
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| − | | |
| − | <math>\textbf{(E)} \: x+y+z=1</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 7|Solution]] | |
| | | | |
| | ==Problem 8== | | ==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?
| + | [[2024 AMC 12B Problems/Problem 8|Solution]] |
| − | | |
| − | <math>\textbf{(A)} \: 105 \qquad\textbf{(B)} \: 120 \qquad\textbf{(C)} \: 135 \qquad\textbf{(D)} \: 150 \qquad\textbf{(E)} \: 165</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 8|Solution]] | |
| | | | |
| | ==Problem 9== | | ==Problem 9== |
| − | | + | [[2024 AMC 12B Problems/Problem 9|Solution]] |
| − | Triangle <math>ABC</math> is equilateral with side length <math>6</math>. Suppose that <math>O</math> is the center of the inscribed
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| − | circle of this triangle. What is the area of the circle passing through <math>A</math>, <math>O</math>, and <math>C</math>?
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| − | | |
| − | <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>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 9|Solution]] | |
| | | | |
| | ==Problem 10== | | ==Problem 10== |
| − | | + | [[2024 AMC 12B Problems/Problem 10|Solution]] |
| − | 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>
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| − | is isosceles?
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| − | | |
| − | <math>\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 10|Solution]] | |
| | | | |
| | ==Problem 11== | | ==Problem 11== |
| − | Una rolls <math>6</math> standard <math>6</math>-sided dice simultaneously and calculates the product of the <math>6{ }</math> numbers obtained. What is the probability that the product is divisible by <math>4?</math>
| + | [[2024 AMC 12B Problems/Problem 11|Solution]] |
| − | | |
| − | <math>\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}</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 11|Solution]] | |
| | | | |
| | ==Problem 12== | | ==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>
| + | [[2024 AMC 12B Problems/Problem 12|Solution]] |
| − | What is <math>f(768)-f(384)?</math>
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| − | | |
| − | <math>\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\
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| − | \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 12|Solution]] | |
| | | | |
| | ==Problem 13== | | ==Problem 13== |
| − | Let <math>c = \frac{2\pi}{11}.</math> What is the value of
| + | [[2024 AMC 12B Problems/Problem 13|Solution]] |
| − | <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>
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| − | | |
| − | <math>\textbf{(A)}\ {-}1 \qquad\textbf{(B)}\ {-}\frac{\sqrt{11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\
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| − | \frac{10}{11} \qquad\textbf{(E)}\ 1</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 13|Solution]] | |
| | | | |
| | ==Problem 14== | | ==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>?
| + | [[2024 AMC 12B Problems/Problem 14|Solution]] |
| − | | |
| − | <math>\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 14|Solution]] | |
| | | | |
| | ==Problem 15== | | ==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>?
| + | [[2024 AMC 12B Problems/Problem 15|Solution]] |
| − | <center><asy>
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| − | defaultpen(fontsize(8)+0.8); size(150);
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| − | pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4;
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| − | real x=45, y=90, z=60; O=origin;
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| − | A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y);
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| − | B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z);
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| − | C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z);
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| − | draw(A1--A2--A3--A4--A1, gray+0.25+dashed);
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| − | filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25));
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| − | filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25));
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| − | dot(O);
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| − | pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4;
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| − | P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4);
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| − | P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1);
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| − | P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2);
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| − | P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3);
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| − | R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4);
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| − | R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2);
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| − | draw(A1--P1--B2--R1--C3--Q1--A2);
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| − | draw(A2--P2--B3--R2--C4--Q2--A3);
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| − | draw(A3--P3--B4--R3--C1--Q3--A4);
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| − | draw(A4--P4--B1--R4--C2--Q4--A1);
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| − | </asy></center>
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| − | <math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 15|Solution]] | |
| | | | |
| | ==Problem 16== | | ==Problem 16== |
| − | | + | [[2024 AMC 12B Problems/Problem 16|Solution]] |
| − | 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>?
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| − | <math>\textbf{(A)}\: 259\qquad\textbf{(B)} \: 438\qquad\textbf{(C)} \: 516\qquad\textbf{(D)} \: 625\qquad\textbf{(E)} \: 687</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 16|Solution]] | |
| | | | |
| | ==Problem 17== | | ==Problem 17== |
| − | A bug starts at a vertex of a grid made of equilateral triangles of side length <math>1</math>. At each step the bug moves in one of the <math>6</math> possible directions along the grid lines randomly and independently with equal probability. What is the probability that after <math>5</math> moves the bug never will have been more than <math>1</math> unit away from the starting position?
| + | [[2024 AMC 12B Problems/Problem 17|Solution]] |
| − | | |
| − | <math>\textbf{(A)}\ \frac{13}{108} \qquad\textbf{(B)}\ \frac{7}{54} \qquad\textbf{(C)}\ \frac{29}{216} \qquad\textbf{(D)}\
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| − | \frac{4}{27} \qquad\textbf{(E)}\ \frac{1}{16}</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 17|Solution]] | |
| | | | |
| | ==Problem 18== | | ==Problem 18== |
| − | | + | [[2024 AMC 12B Problems/Problem 18|Solution]] |
| − | 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>
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| − | | |
| − | 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>
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| − | <math>\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 18|Solution]] | |
| | | | |
| | ==Problem 19== | | ==Problem 19== |
| − | 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?
| + | [[2024 AMC 12B Problems/Problem 19|Solution]] |
| − | | |
| − | <math>(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 19|Solution]] | |
| | | | |
| | ==Problem 20== | | ==Problem 20== |
| − | A cube is constructed from <math>4</math> white unit cubes and <math>4</math> blue unit cubes. How many different ways are there to construct the <math>2 \times 2 \times 2</math> cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
| + | [[2024 AMC 12B Problems/Problem 20|Solution]] |
| − | | |
| − | <math>(\textbf{A})\: 7\qquad(\textbf{B}) \: 8\qquad(\textbf{C}) \: 9\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 11</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 20|Solution]] | |
| | | | |
| | ==Problem 21== | | ==Problem 21== |
| − | | + | [[2024 AMC 12B Problems/Problem 21|Solution]] |
| − | For real numbers <math>x</math>, let
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| − | <cmath>P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)</cmath>
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| − | where <math>i = \sqrt{-1}</math>. For how many values of <math>x</math> with <math>0\leq x<2\pi</math> does
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| − | <cmath>P(x)=0?</cmath>
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| − | | |
| − | <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\
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| − | 3 \qquad\textbf{(E)}\ 4</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 21|Solution]] | |
| | | | |
| | ==Problem 22== | | ==Problem 22== |
| − | | + | [[2024 AMC 12B Problems/Problem 22|Solution]] |
| − | 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>?
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| − | | |
| − | <math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\
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| − | \frac{73}{25} \qquad\textbf{(E)}\ 3</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 22|Solution]] | |
| | | | |
| | ==Problem 23== | | ==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.)
| + | [[2024 AMC 12B Problems/Problem 23|Solution]] |
| − | | |
| − | <math>\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
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| − | \frac{29}{30} \qquad\textbf{(E)}\ 1</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 23|Solution]] | |
| | | | |
| | ==Problem 24== | | ==Problem 24== |
| − | | + | [[2024 AMC 12B Problems/Problem 24|Solution]] |
| − | 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>?
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| − | | |
| − | <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>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 24|Solution]] | |
| | | | |
| | ==Problem 25== | | ==Problem 25== |
| | + | [[2024 AMC 12B Problems/Problem 25|Solution]] |
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| − | 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>
| + | [[2024 AMC 12B Problems|Back to the list of problems]] |
| − | | |
| − | <math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4</math>
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| − | | |
| − | [[2021 Fall AMC 12B Problems/Problem 25|Solution]] | |
| − | | |
| − | ==See also==
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| − | {{AMC12 box|year=2021 Fall|ab=B|before=[[2021 Fall AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}}
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| − | {{MAA Notice}}
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