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Difference between revisions of "2022 AMC 12B Problems"

m (→‎Problem 9)
Line 5: Line 5:
 
Define <math>x\diamond y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. What is the value of <cmath>(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?</cmath>
 
Define <math>x\diamond y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. What is the value of <cmath>(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?</cmath>
  
−
<math> \textbf{(A)}\ -2 \qquad
+
<math> \textbf{(A) }\ -2 \qquad
−
\textbf{(B)}\ -1 \qquad
+
\textbf{(B) }\ -1 \qquad
−
\textbf{(C)}\ 0 \qquad
+
\textbf{(C) }\ 0 \qquad
−
\textbf{(D)}\ 1 \qquad
+
\textbf{(D) }\ 1 \qquad
−
\textbf{(E)}\ 2</math>
+
\textbf{(E) }\ 2</math>
  
 
[[2022 AMC 12B Problems/Problem 1|Solution]]
 
[[2022 AMC 12B Problems/Problem 1|Solution]]
Line 43: Line 43:
 
How many of the first ten numbers of the sequence <math>121</math>, <math>11211</math>, <math>1112111</math>, ... are prime numbers?
 
How many of the first ten numbers of the sequence <math>121</math>, <math>11211</math>, <math>1112111</math>, ... are prime numbers?
  
−
<math>\text{(A) } 0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }3 \qquad \text{(E) }4</math>
+
<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math>
  
 
[[2022 AMC 12B Problems/Problem 3|Solution]]
 
[[2022 AMC 12B Problems/Problem 3|Solution]]
Line 59: Line 59:
 
The point <math>(-1, -2)</math> is rotated <math>270^{\circ}</math> counterclockwise about the point <math>(3, 1)</math>. What are the coordinates of its new position?
 
The point <math>(-1, -2)</math> is rotated <math>270^{\circ}</math> counterclockwise about the point <math>(3, 1)</math>. What are the coordinates of its new position?
  
−
<math>\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)</math>
+
<math>\textbf{(A) }\ (-3, -4) \qquad \textbf{(B) }\ (0,5) \qquad \textbf{(C) }\ (2,-1) \qquad \textbf{(D) }\ (4,3) \qquad \textbf{(E) }\ (6,-3)</math>
  
 
[[2022 AMC 12B Problems/Problem 5|Solution]]
 
[[2022 AMC 12B Problems/Problem 5|Solution]]
Line 75: Line 75:
 
How many of these sets contain exactly two multiples of <math>7</math>?
 
How many of these sets contain exactly two multiples of <math>7</math>?
  
−
<math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50</math>
+
<math>\textbf{(A) }\ 40\qquad\textbf{(B) }\ 42\qquad\textbf{(C) }\ 43\qquad\textbf{(D) }\ 49\qquad\textbf{(E) }\ 50</math>
  
 
[[2022 AMC 12B Problems/Problem 6|Solution]]
 
[[2022 AMC 12B Problems/Problem 6|Solution]]
Line 91: Line 91:
 
What is the graph of <math>y^4+1=x^4+2y^2</math> in the coordinate plane?
 
What is the graph of <math>y^4+1=x^4+2y^2</math> in the coordinate plane?
  
−
<math>\textbf{(A)}\ \text{two intersecting parabolas} \qquad \textbf{(B)}\ \text{two nonintersecting parabolas} \qquad \textbf{(C)}\ \text{two intersecting circles} \qquad</math>
+
<math>\textbf{(A) }\ \text{two intersecting parabolas} \qquad \textbf{(B) }\ \text{two nonintersecting parabolas} \qquad \textbf{(C) }\ \text{two intersecting circles} \qquad</math>
  
−
<math>\textbf{(D)}\ \text{a circle and a hyperbola} \qquad \textbf{(E)}\ \text{a circle and two parabolas}</math>
+
<math>\textbf{(D) }\ \text{a circle and a hyperbola} \qquad \textbf{(E) }\ \text{a circle and two parabolas}</math>
  
 
[[2022 AMC 12B Problems/Problem 8|Solution]]
 
[[2022 AMC 12B Problems/Problem 8|Solution]]
Line 101: Line 101:
 
The sequence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> What is the minimum possible value of <math>a_2</math>?
 
The sequence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> What is the minimum possible value of <math>a_2</math>?
  
−
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 22</math>
+
<math>\textbf{(A) }\ 8 \qquad \textbf{(B) }\ 12 \qquad \textbf{(C) }\ 16 \qquad \textbf{(D) }\ 17 \qquad \textbf{(E) }\ 22</math>
  
 
[[2022 AMC 12B Problems/Problem 9|Solution]]
 
[[2022 AMC 12B Problems/Problem 9|Solution]]
Line 109: Line 109:
 
Regular hexagon <math>ABCDEF</math> has side length <math>2</math>. Let <math>G</math> be the midpoint of <math>\overline{AB}</math>, and let <math>H</math> be the midpoint of <math>\overline{DE}</math>. What is the perimeter of <math>GCHF</math>?
 
Regular hexagon <math>ABCDEF</math> has side length <math>2</math>. Let <math>G</math> be the midpoint of <math>\overline{AB}</math>, and let <math>H</math> be the midpoint of <math>\overline{DE}</math>. What is the perimeter of <math>GCHF</math>?
  
−
<math> \textbf{(A)}\ 4\sqrt3 \qquad
+
<math> \textbf{(A) }\ 4\sqrt3 \qquad
−
\textbf{(B)}\ 8 \qquad
+
\textbf{(B) }\ 8 \qquad
−
\textbf{(C)}\ 4\sqrt5 \qquad
+
\textbf{(C) }\ 4\sqrt5 \qquad
−
\textbf{(D)}\ 4\sqrt7 \qquad
+
\textbf{(D) }\ 4\sqrt7 \qquad
−
\textbf{(E)}\ 12</math>
+
\textbf{(E) }\ 12</math>
  
 
[[2022 AMC 12B Problems/Problem 10|Solution]]
 
[[2022 AMC 12B Problems/Problem 10|Solution]]
Line 121: Line 121:
 
Let <math> f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n </math>, where <math>i = \sqrt{-1}</math>. What is <math>f(2022)</math>?
 
Let <math> f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n </math>, where <math>i = \sqrt{-1}</math>. What is <math>f(2022)</math>?
  
−
<math> \textbf{(A)}\ -2 \qquad
+
<math> \textbf{(A) }\ -2 \qquad
−
\textbf{(B)}\ -1 \qquad
+
\textbf{(B) }\ -1 \qquad
−
\textbf{(C)}\ 0 \qquad
+
\textbf{(C) }\ 0 \qquad
−
\textbf{(D)}\ \sqrt{3} \qquad
+
\textbf{(D) }\ \sqrt{3} \qquad
−
\textbf{(E)}\ 2</math>
+
\textbf{(E) }\ 2</math>
  
 
[[2022 AMC 12B Problems/Problem 11|Solution]]
 
[[2022 AMC 12B Problems/Problem 11|Solution]]
Line 131: Line 131:
 
==Problem 12 ==
 
==Problem 12 ==
  
−
XXX
+
Kayla rolls four fair <math>6</math>-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than <math>4</math> and at least two of the numbers she rolls are greater than <math>2</math>?
 +
 
 +
<math>\textbf{(A) }\frac{2}{3} \qquad \textbf{(B) }\frac{19}{27} \qquad \textbf{(C) }\frac{59}{81} \qquad \textbf{(D) }\frac{61}{81} \qquad \textbf{(E) }\frac{7}{9}</math>
  
 
[[2022 AMC 12B Problems/Problem 12|Solution]]
 
[[2022 AMC 12B Problems/Problem 12|Solution]]
Line 137: Line 139:
 
==Problem 13 ==
 
==Problem 13 ==
  
−
XXX
+
The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
 +
 
 +
<asy>
 +
        import geometry;
 +
        unitsize(0.75cm);
 +
 
 +
        draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.5));
 +
        draw((1,4)--(4,0)--(8,3)--(5,7)--cycle,black+linewidth(1.5));
 +
        filldraw((1,4)--(4,0)--(8,3)--(29/4,4)--cycle,gray+opacity(0.5),invisible);
 +
        draw((1,0)--(1,4),linewidth(1.5));
 +
        perpendicularmark((1,0),unit(dir(90)+dir(0)),black+linewidth(1.5));
 +
        label("\Large8",(4,-0.5),S);
 +
        label("\Large4",(8.5,2),E);
 +
        label("\Large5",(3,5.5),NW);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }15\dfrac{1}{8}  \qquad
 +
\textbf{(B) }15\dfrac{3}{8}  \qquad
 +
\textbf{(C) }15\dfrac{1}{2}  \qquad
 +
\textbf{(D) }15\dfrac{5}{8}  \qquad
 +
\textbf{(E) }15\dfrac{7}{8} </math>
  
 
[[2022 AMC 12B Problems/Problem 13|Solution]]
 
[[2022 AMC 12B Problems/Problem 13|Solution]]
Line 143: Line 165:
 
==Problem 14 ==
 
==Problem 14 ==
  
−
XXX
+
The graph of <math>y=x^2+2x-15</math> intersects the <math>x</math>-axis at points <math>A</math> and <math>C</math> and the <math>y</math>-axis at point <math>B</math>. What is <math>\tan(\angle ABC)</math>?
 +
 
 +
<math>\textbf{(A) }\frac{1}{7} \qquad \textbf{(B) }\frac{1}{4} \qquad \textbf{(C) }\frac{3}{7} \qquad \textbf{(D) }\frac{1}{2} \qquad \textbf{(E) }\frac{4}{7}</math>
  
 
[[2022 AMC 12B Problems/Problem 14|Solution]]
 
[[2022 AMC 12B Problems/Problem 14|Solution]]
Line 149: Line 173:
 
==Problem 15 ==
 
==Problem 15 ==
  
−
XXX
+
One of the following numbers is not divisible by any prime number less than 10. Which is it?
 +
<math>\textbf{(A) } 2^{606} - 1 \qquad  \textbf{(B) } 2^{606} + 1 \qquad  \textbf{(C) } 2^{607} - 1 \qquad  \textbf{(D) } 2^{607} + 1 \qquad  \textbf{(E) } 2^{607} + 3^{607}</math>
  
 
[[2022 AMC 12B Problems/Problem 15|Solution]]
 
[[2022 AMC 12B Problems/Problem 15|Solution]]
Line 155: Line 180:
 
==Problem 16 ==
 
==Problem 16 ==
  
−
XXX
+
Suppose <math>x</math> and <math>y</math> are positive real numbers such that
 +
 
 +
<math>x^y=2^{64}</math> and <math>(\log_2{x})^{\log_2{y}}=2^{27}</math>.
 +
 
 +
What is the greatest possible value of <math>\log_2{y}</math>?
 +
 
 +
<math>\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }3+\sqrt{2} \qquad \textbf{(D) }4+\sqrt{3} \qquad \textbf{(E) }7</math>
  
 
[[2022 AMC 12B Problems/Problem 16|Solution]]
 
[[2022 AMC 12B Problems/Problem 16|Solution]]
Line 162: Line 193:
 
How many <math>4 \times 4</math> arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in each column) are also 1, 2, 3, and 4, in some order? For example, the array<cmath> \left[  \begin{array}{cccc}    1 & 1 & 1 & 0 \\    0 & 1 & 1 & 0 \\    1 & 1 & 1 & 1 \\    0 & 1 & 0 & 0 \\  \end{array} \right] </cmath>
 
How many <math>4 \times 4</math> arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in each column) are also 1, 2, 3, and 4, in some order? For example, the array<cmath> \left[  \begin{array}{cccc}    1 & 1 & 1 & 0 \\    0 & 1 & 1 & 0 \\    1 & 1 & 1 & 1 \\    0 & 1 & 0 & 0 \\  \end{array} \right] </cmath>
 
satisfies the condition.
 
satisfies the condition.
 +
 +
<math>\textbf{(A) }144 \qquad \textbf{(B) }240 \qquad \textbf{(C) }336 \qquad \textbf{(D) }576 \qquad \textbf{(E) }624</math>
  
 
[[2022 AMC 12B Problems/Problem 17|Solution]]
 
[[2022 AMC 12B Problems/Problem 17|Solution]]
Line 167: Line 200:
 
==Problem 18 ==
 
==Problem 18 ==
  
−
XXX
+
Each square in a <math>5 \times 5</math> grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
 +
Any filled square with two or three filled neighbors remains filled.
 +
Any empty square with exactly three filled neighbors becomes a filled square.
 +
All other squares remain empty or become empty.
 +
A sample transformation is shown in the figure below.
 +
 
 +
<asy>
 +
        import geometry;
 +
        unitsize(0.6cm);
 +
 
 +
        void ds(pair x) {
 +
            filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
 +
        }
 +
 
 +
        ds((1,1));
 +
        ds((2,1));
 +
        ds((3,1));
 +
        ds((1,3));
 +
 
 +
        for (int i = 0; i <= 5; ++i) {
 +
            draw((0,i)--(5,i));
 +
            draw((i,0)--(i,5));
 +
        }
 +
 
 +
        label("Initial", (2.5,-1));
 +
        draw((6,2.5)--(8,2.5),Arrow);
 +
 
 +
        ds((10,2));
 +
        ds((11,1));
 +
        ds((11,0));
 +
 
 +
        for (int i = 0; i <= 5; ++i) {
 +
            draw((9,i)--(14,i));
 +
            draw((i+9,0)--(i+9,5));
 +
        }
 +
 
 +
        label("Transformed", (11.5,-1));
 +
</asy>
 +
 
 +
Suppose the <math>5 \times 5</math> grid has a border of empty squares surrounding a <math>3 \times 3</math> subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
 +
<asy>
 +
        import geometry;
 +
        unitsize(0.6cm);
 +
 
 +
        void ds(pair x) {
 +
            filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible);
 +
        }
 +
 
 +
        for (int i = 1; i < 4; ++ i) {
 +
            for (int j = 1; j < 4; ++j) {
 +
                label("?",(i + 0.5, j + 0.5));
 +
            }
 +
        }
 +
 
 +
        for (int i = 0; i <= 5; ++i) {
 +
            draw((0,i)--(5,i));
 +
            draw((i,0)--(i,5));
 +
        }
 +
 
 +
        label("Initial", (2.5,-1));
 +
        draw((6,2.5)--(8,2.5),Arrow);
 +
 
 +
        ds((11,2));
 +
 
 +
        for (int i = 0; i <= 5; ++i) {
 +
            draw((9,i)--(14,i));
 +
            draw((i+9,0)--(i+9,5));
 +
        }
 +
 
 +
        label("Transformed", (11.5,-1));
 +
</asy>
 +
 
 +
<math>\textbf{(A) } 14 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 26 \qquad \textbf{(E) } 30</math>
  
 
[[2022 AMC 12B Problems/Problem 18|Solution]]
 
[[2022 AMC 12B Problems/Problem 18|Solution]]
Line 173: Line 278:
 
==Problem 19 ==
 
==Problem 19 ==
  
−
XXX
+
In <math>\triangle{ABC}</math> medians <math>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>G</math> and <math>\triangle{AGE}</math> is equilateral. Then <math>\cos(C)</math> can be written as <math>\frac{m\sqrt p}n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>p</math> is a positive integer not divisible by the square of any prime. What is <math>m+n+p?</math>
 +
<asy>
 +
            import geometry;
 +
            unitsize(2cm);
 +
 
 +
real arg(pair p) {
 +
              return atan2(p.y, p.x) * 180/pi;
 +
            }
 +
 
 +
            pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C;
 +
 
 +
            pair t(pair p) {
 +
                return rotate(-arg(dir(B--C)))*p;
 +
            }
 +
 
 +
 
 +
            path t(path p) {
 +
                return rotate(-arg(dir(B--C)))*p;
 +
            }
 +
 
 +
            void d(path p, pen q = black+linewidth(1.5)) {
 +
                draw(t(p),q);
 +
            }
 +
 
 +
            void o(pair p, pen q = 5+black) {
 +
                dot(t(p),q);
 +
            }
 +
 
 +
            void l(string s, pair p, pair d) {
 +
                label(s, t(p),d);
 +
            }
 +
           
 +
            d(A--B--C--cycle);
 +
            d(A--D);
 +
            d(B--E);
 +
            o(A);
 +
            o(B);
 +
            o(C);
 +
            o(D);
 +
            o(E);
 +
            o(G);
 +
            l("$A$",A,N);
 +
            l("$B$",B,SW);
 +
            l("$C$",C,SE);
 +
            l("$D$",D,S);
 +
            l("$E$",E,NE);
 +
            l("$G$",G,NW);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60</math>
  
 
[[2022 AMC 12B Problems/Problem 19|Solution]]
 
[[2022 AMC 12B Problems/Problem 19|Solution]]
Line 179: Line 333:
 
==Problem 20 ==
 
==Problem 20 ==
  
−
XXX
+
Let <math>P(x)</math> be a polynomial with rational coefficients such that when <math>P(x)</math> is divided by the polynomial <math>x^2 + x + 1</math>, the remainder is <math>x + 2</math>, and when <math>P(x)</math> is divided by the polynomial <math>x^2 + 1</math>, the remainder is <math>2x + 1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
 +
 
 +
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23</math>
  
 
[[2022 AMC 12B Problems/Problem 20|Solution]]
 
[[2022 AMC 12B Problems/Problem 20|Solution]]
  
 +
==Problem 21 ==
  
−
==Problem 21 ==
+
Let <math>S</math> be the set of circles in the coordinate plane that are tangent to each of the three circles with equations <math>x^{2}+y^{2}=4</math>, <math>x^{2}+y^{2}=64</math>, and <math>(x-5)^{2}+y^{2}=3</math>. What is the sum of the areas of all circles in <math>S</math>?
  
−
XXX
+
<math>\textbf{(A) } 48 \pi \qquad
 +
\textbf{(B) } 68 \pi \qquad
 +
\textbf{(C) } 96 \pi \qquad
 +
\textbf{(D) } 102 \pi \qquad
 +
\textbf{(E) } 136 \pi \qquad</math>
  
 
[[2022 AMC 12B Problems/Problem 21|Solution]]
 
[[2022 AMC 12B Problems/Problem 21|Solution]]
Line 192: Line 353:
 
==Problem 22 ==
 
==Problem 22 ==
  
−
XXX
+
Ant Amelia starts on the number line at <math>0</math> and crawls in the following manner. For <math>n=1,2,3,</math> Amelia chooses a time duration <math>t_n</math> and an increment <math>x_n</math> independently and uniformly at random from the interval <math>(0,1).</math> During the <math>n</math>th step of the process, Amelia moves <math>x_n</math> units in the positive direction, using up <math>t_n</math> minutes. If the total elapsed time has exceeded <math>1</math> minute during the <math>n</math>th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most <math>3</math> steps in all. What is the probability that Amelia’s position when she stops will be greater than <math>1</math>?
 +
 
 +
<math>\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}</math>
  
 
[[2022 AMC 12B Problems/Problem 22|Solution]]
 
[[2022 AMC 12B Problems/Problem 22|Solution]]
Line 204: Line 367:
  
  
−
<math>\textbf{(A)}~6\qquad\textbf{(B)}~7\qquad\textbf{(C)}~12\qquad\textbf{(D)}~14\qquad\textbf{(E)}~15\qquad</math>
+
<math>\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) }12\qquad \textbf{(D) } 14\qquad \textbf{(E) }15</math>
  
  
Line 211: Line 374:
 
==Problem 24 ==
 
==Problem 24 ==
  
−
XXX
+
The figure below depicts a regular 7-gon inscribed in a unit circle.
 +
<asy>
 +
        import geometry;
 +
unitsize(3cm);
 +
draw(circle((0,0),1),linewidth(1.5));
 +
for (int i = 0; i < 7; ++i) {
 +
  for (int j = 0; j < i; ++j) {
 +
    draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5));
 +
  }
 +
}
 +
for(int i = 0; i < 7; ++i) {
 +
  dot(dir(i * 360/7),5+black);
 +
}
 +
</asy>
 +
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?
 +
 
 +
<math>\textbf{(A) }49 \qquad \textbf{(B) }98 \qquad \textbf{(C) }147 \qquad \textbf{(D) }168 \qquad \textbf{(E) }196</math>
  
 
[[2022 AMC 12B Problems/Problem 24|Solution]]
 
[[2022 AMC 12B Problems/Problem 24|Solution]]
Line 217: Line 396:
 
==Problem 25 ==
 
==Problem 25 ==
  
−
XXX
+
Four regular hexagons surround a square with a side length <math>1</math>, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as <math>m\sqrt{n} + p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are integers and <math>n</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?
 +
 
 +
<asy>
 +
        import geometry;
 +
        unitsize(3cm);
 +
        draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
 +
        draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
 +
        draw(shift((1/2,sqrt(3)/2))*polygon(6));
 +
        draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
 +
        draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6));
 +
draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2));
 +
</asy>
 +
 
 +
<math>\textbf{(A) } -12 \qquad
 +
\textbf{(B) }-4 \qquad
 +
\textbf{(C) } 4 \qquad
 +
\textbf{(D) }24 \qquad
 +
\textbf{(E) }32</math>
  
 
[[2022 AMC 12B Problems/Problem 25|Solution]]
 
[[2022 AMC 12B Problems/Problem 25|Solution]]

Revision as of 22:36, 17 November 2022

2022 AMC 12B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. 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.
  3. 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).
  4. Figures are not necessarily drawn to scale.
  5. 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

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]

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

Solution

Problem 2

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? (Note: The figure is not drawn to scale.)

[asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/asy]

$\textbf{(A) }3\sqrt 5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }6\sqrt 5 \qquad \textbf{(D) }20\qquad \textbf{(E) }25$

Solution

Problem 3

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?

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

Solution

Problem 4

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

Solution

Problem 5

The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?

$\textbf{(A) }\ (-3, -4) \qquad \textbf{(B) }\ (0,5) \qquad \textbf{(C) }\ (2,-1) \qquad \textbf{(D) }\ (4,3) \qquad \textbf{(E) }\ (6,-3)$

Solution

Problem 6

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\cdots,10\}, \\ &\{11,12,13,\cdots,20\},\\ &\{21,22,23,\cdots,30\},\\ &\vdots\\ &\{991,992,993,\cdots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$?

$\textbf{(A) }\ 40\qquad\textbf{(B) }\ 42\qquad\textbf{(C) }\ 43\qquad\textbf{(D) }\ 49\qquad\textbf{(E) }\ 50$

Solution

Problem 7

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$

Solution

Problem 8

What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?

$\textbf{(A) }\ \text{two intersecting parabolas} \qquad \textbf{(B) }\ \text{two nonintersecting parabolas} \qquad \textbf{(C) }\ \text{two intersecting circles} \qquad$

$\textbf{(D) }\ \text{a circle and a hyperbola} \qquad \textbf{(E) }\ \text{a circle and two parabolas}$

Solution

Problem 9

The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that \[2^{a_7}=2^{27} \cdot a_7.\] What is the minimum possible value of $a_2$?

$\textbf{(A) }\ 8 \qquad \textbf{(B) }\ 12 \qquad \textbf{(C) }\ 16 \qquad \textbf{(D) }\ 17 \qquad \textbf{(E) }\ 22$

Solution

Problem 10

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?

$\textbf{(A) }\ 4\sqrt3 \qquad \textbf{(B) }\ 8 \qquad \textbf{(C) }\ 4\sqrt5 \qquad \textbf{(D) }\ 4\sqrt7 \qquad \textbf{(E) }\ 12$

Solution

Problem 11

Let $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$, where $i = \sqrt{-1}$. What is $f(2022)$?

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

Solution

Problem 12

Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?

$\textbf{(A) }\frac{2}{3} \qquad \textbf{(B) }\frac{19}{27} \qquad \textbf{(C) }\frac{59}{81} \qquad \textbf{(D) }\frac{61}{81} \qquad \textbf{(E) }\frac{7}{9}$

Solution

Problem 13

The diagram below shows a rectangle with side lengths 4 and 8 and a square with side length 5. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

[asy] import geometry; unitsize(0.75cm); draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.5)); draw((1,4)--(4,0)--(8,3)--(5,7)--cycle,black+linewidth(1.5)); filldraw((1,4)--(4,0)--(8,3)--(29/4,4)--cycle,gray+opacity(0.5),invisible); draw((1,0)--(1,4),linewidth(1.5)); perpendicularmark((1,0),unit(dir(90)+dir(0)),black+linewidth(1.5)); label("\Large8",(4,-0.5),S); label("\Large4",(8.5,2),E); label("\Large5",(3,5.5),NW); [/asy]

$\textbf{(A) }15\dfrac{1}{8} \qquad \textbf{(B) }15\dfrac{3}{8} \qquad \textbf{(C) }15\dfrac{1}{2} \qquad \textbf{(D) }15\dfrac{5}{8} \qquad \textbf{(E) }15\dfrac{7}{8}$

Solution

Problem 14

The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?

$\textbf{(A) }\frac{1}{7} \qquad \textbf{(B) }\frac{1}{4} \qquad \textbf{(C) }\frac{3}{7} \qquad \textbf{(D) }\frac{1}{2} \qquad \textbf{(E) }\frac{4}{7}$

Solution

Problem 15

One of the following numbers is not divisible by any prime number less than 10. Which is it? $\textbf{(A) } 2^{606} - 1 \qquad \textbf{(B) } 2^{606} + 1 \qquad \textbf{(C) } 2^{607} - 1 \qquad \textbf{(D) } 2^{607} + 1 \qquad \textbf{(E) } 2^{607} + 3^{607}$

Solution

Problem 16

Suppose $x$ and $y$ are positive real numbers such that

$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{27}$.

What is the greatest possible value of $\log_2{y}$?

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }3+\sqrt{2} \qquad \textbf{(D) }4+\sqrt{3} \qquad \textbf{(E) }7$

Solution

Problem 17

How many $4 \times 4$ arrays whose entries are 0s and 1s are there such that the row sums (the sum of the entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in each column) are also 1, 2, 3, and 4, in some order? For example, the array\[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]\] satisfies the condition.

$\textbf{(A) }144 \qquad \textbf{(B) }240 \qquad \textbf{(C) }336 \qquad \textbf{(D) }576 \qquad \textbf{(E) }624$

Solution

Problem 18

Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: Any filled square with two or three filled neighbors remains filled. Any empty square with exactly three filled neighbors becomes a filled square. All other squares remain empty or become empty. A sample transformation is shown in the figure below.

[asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } ds((1,1)); ds((2,1)); ds((3,1)); ds((1,3)); for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((10,2)); ds((11,1)); ds((11,0)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy]

Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.) [asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } for (int i = 1; i < 4; ++ i) { for (int j = 1; j < 4; ++j) { label("?",(i + 0.5, j + 0.5)); } } for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((11,2)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy]

$\textbf{(A) } 14 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 26 \qquad \textbf{(E) } 30$

Solution

Problem 19

In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$ [asy] import geometry; unitsize(2cm); real arg(pair p) { return atan2(p.y, p.x) * 180/pi; } pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C; pair t(pair p) { return rotate(-arg(dir(B--C)))*p; } path t(path p) { return rotate(-arg(dir(B--C)))*p; } void d(path p, pen q = black+linewidth(1.5)) { draw(t(p),q); } void o(pair p, pen q = 5+black) { dot(t(p),q); } void l(string s, pair p, pair d) { label(s, t(p),d); } d(A--B--C--cycle); d(A--D); d(B--E); o(A); o(B); o(C); o(D); o(E); o(G); l("$A$",A,N); l("$B$",B,SW); l("$C$",C,SE); l("$D$",D,S); l("$E$",E,NE); l("$G$",G,NW); [/asy]

$\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60$

Solution

Problem 20

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$

Solution

Problem 21

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?

$\textbf{(A) } 48 \pi \qquad \textbf{(B) } 68 \pi \qquad \textbf{(C) } 96 \pi \qquad \textbf{(D) } 102 \pi \qquad \textbf{(E) } 136 \pi \qquad$

Solution

Problem 22

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?

$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$

Solution

Problem 23

Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \[S_n = \sum_{k=0}^{n-1} x_k 2^k\]

Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geqslant 1$. What is the value of the sum \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}\]


$\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) }12\qquad \textbf{(D) } 14\qquad \textbf{(E) }15$


Solution

Problem 24

The figure below depicts a regular 7-gon inscribed in a unit circle. [asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy] What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?

$\textbf{(A) }49 \qquad \textbf{(B) }98 \qquad \textbf{(C) }147 \qquad \textbf{(D) }168 \qquad \textbf{(E) }196$

Solution

Problem 25

Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?

[asy] import geometry; unitsize(3cm); draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle); draw(shift((1/2,1-sqrt(3)/2))*polygon(6)); draw(shift((1/2,sqrt(3)/2))*polygon(6)); draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2)); [/asy]

$\textbf{(A) } -12 \qquad \textbf{(B) }-4 \qquad \textbf{(C) } 4 \qquad \textbf{(D) }24 \qquad \textbf{(E) }32$

Solution

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