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Difference between revisions of "2022 AMC 10B Problems"

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==Problem 1 ==
 
==Problem 1 ==
−
XXX
 
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
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
 +
\textbf{(B)}\ {-}1 \qquad
 +
\textbf{(C)}\ 0 \qquad
 +
\textbf{(D)}\ 1 \qquad
 +
\textbf{(E)}\ 2</math>
 +
 
  
 
[[2022 AMC 10B Problems/Problem 1|Solution]]
 
[[2022 AMC 10B Problems/Problem 1|Solution]]
 +
 
==Problem 2 ==
 
==Problem 2 ==
−
XXX
 
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
In rhombus <math>ABCD</math>, point <math>P</math> lies on segment <math>\overline{AD}</math> so that <math>\overline{BP}</math> <math>\perp</math> <math>\overline{AD}</math>, <math>AP = 3</math>, and <math>PD = 2</math>. What is the area of <math>ABCD</math>? (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>
 +
 
 +
<math>\textbf{(A) }3\sqrt 5 \qquad
 +
\textbf{(B) }10 \qquad
 +
\textbf{(C) }6\sqrt 5 \qquad
 +
\textbf{(D) }20\qquad
 +
\textbf{(E) }25</math>
  
 
[[2022 AMC 10B Problems/Problem 2|Solution]]
 
[[2022 AMC 10B Problems/Problem 2|Solution]]
 +
 
==Problem 3 ==
 
==Problem 3 ==
 
How many three-digit positive integers have an odd number of even digits?
 
How many three-digit positive integers have an odd number of even digits?
Line 21: Line 48:
  
 
==Problem 4 ==
 
==Problem 4 ==
−
A donkey suffers an attack of hiccups and the first hiccup happens at <math>\text{4:00}</math> one afternoon. Suppose that
+
A donkey suffers an attack of hiccups and the first hiccup happens at <math>4:00</math> one afternoon. Suppose that
−
the donkey hiccups regularly every <math>5</math> seconds. At what time does the donkey’s <math>\text{700th}</math> hiccup occur?
+
the donkey hiccups regularly every <math>5</math> seconds. At what time does the donkey’s <math>700</math>th hiccup occur?
  
−
<math>\textbf{(A) }</math> <math>15</math> seconds after <math>\text{4:58}</math>
+
<math>\textbf{(A) }15 \text{ seconds after } 4:58</math>
  
−
<math>\textbf{(B) }</math> <math>20</math> seconds after <math>\text{4:58}</math>
+
<math>\textbf{(B) }20 \text{ seconds after } 4:58</math>
  
−
<math>\textbf{(C)}</math> <math>25</math> seconds after <math>\text{4:58}</math>
+
<math>\textbf{(C) }25 \text{ seconds after } 4:58</math>
  
−
<math>\textbf{(D) }</math> <math>30</math> seconds after <math>\text{4:58}</math>
+
<math>\textbf{(D) }30 \text{ seconds after } 4:58</math>
  
−
<math>\textbf{(E) }</math> <math>35</math> seconds after <math>\text{4:58}</math>
+
<math>\textbf{(E) }35 \text{ seconds after } 4:58</math>
  
 
[[2022 AMC 10B Problems/Problem 4|Solution]]
 
[[2022 AMC 10B Problems/Problem 4|Solution]]
  
 
==Problem 5 ==
 
==Problem 5 ==
−
XXX
+
What is the value of <cmath>\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?</cmath>
−
 
+
<math>\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}</math>
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
 
  
 
[[2022 AMC 10B Problems/Problem 5|Solution]]
 
[[2022 AMC 10B Problems/Problem 5|Solution]]
  
 
==Problem 6 ==
 
==Problem 6 ==
−
How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \cdots</math> are prime numbers?
+
How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers?
  
 
<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math>
 
<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math>
Line 51: Line 77:
  
 
==Problem 7 ==
 
==Problem 7 ==
−
XXX
+
For how many values of the constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16</math>
  
 
[[2022 AMC 10B Problems/Problem 7|Solution]]
 
[[2022 AMC 10B Problems/Problem 7|Solution]]
Line 60: Line 86:
 
Consider the following <math>100</math> sets of <math>10</math> elements each:
 
Consider the following <math>100</math> sets of <math>10</math> elements each:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
&\{1,2,3,\cdots,10\}, \\
+
&\{1,2,3,\ldots,10\}, \\
−
&\{11,12,13,\cdots,20\},\\
+
&\{11,12,13,\ldots,20\},\\
−
&\{21,22,23,\cdots,30\},\\
+
&\{21,22,23,\ldots,30\},\\
 
&\vdots\\
 
&\vdots\\
−
&\{991,992,993,\cdots,1000\}.
+
&\{991,992,993,\ldots,1000\}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
 
 
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)} 42\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 10B Problems/Problem 8|Solution]]
 
[[2022 AMC 10B Problems/Problem 8|Solution]]
Line 75: Line 100:
 
==Problem 9 ==
 
==Problem 9 ==
 
The sum
 
The sum
−
<cmath>\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}</cmath>can be expressed as <math>a-\frac{1}{b!}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>?
+
<cmath>\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}</cmath> can be expressed as <math>a-\frac{1}{b!}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>?
  
 
<math> \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024</math>
 
<math> \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024</math>
Line 82: Line 107:
  
 
==Problem 10 ==
 
==Problem 10 ==
−
XXX
+
Camila writes down five positive integers. The unique mode of these integers is <math>2</math> greater than their median, and the median is <math>2</math> greater than their arithmetic mean. What is the least possible value for the mode?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13</math>
  
 
[[2022 AMC 10B Problems/Problem 10|Solution]]
 
[[2022 AMC 10B Problems/Problem 10|Solution]]
  
−
==Problem 11 ==
+
==Problem 11==
−
XXX
+
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
 +
 
 +
<math>\textbf{(A) }</math> All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
 +
 
 +
<math>\textbf{(B) }</math> No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
 +
 
 +
<math>\textbf{(C) }</math> All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
 +
 
 +
<math>\textbf{(D) }</math> All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(E) }</math> All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
  
 
[[2022 AMC 10B Problems/Problem 11|Solution]]
 
[[2022 AMC 10B Problems/Problem 11|Solution]]
Line 103: Line 136:
  
 
==Problem 13 ==
 
==Problem 13 ==
−
XXX
+
The positive difference between a pair of primes is equal to <math>2</math>, and the positive difference between the cubes of the two primes is <math>31106</math>. What is the sum of the digits of the least prime that is greater than those two primes?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 16</math>
  
 
[[2022 AMC 10B Problems/Problem 13|Solution]]
 
[[2022 AMC 10B Problems/Problem 13|Solution]]
  
 
==Problem 14 ==
 
==Problem 14 ==
−
XXX
+
Suppose that <math>S</math> is a subset of <math>\left\{ 1, 2, 3, \ldots , 25 \right\}</math> such that the sum of any two (not necessarily distinct) elements of <math>S</math> is never an element of <math>S.</math> What is the maximum number of elements <math>S</math> may contain?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16</math>
  
 
[[2022 AMC 10B Problems/Problem 14|Solution]]
 
[[2022 AMC 10B Problems/Problem 14|Solution]]
  
 
==Problem 15 ==
 
==Problem 15 ==
−
XXX
+
Let <math>S_n</math> be the sum of the first <math>n</math> terms of an arithmetic sequence that has a common difference of <math>2</math>. The quotient <math>\frac{S_{3n}}{S_n}</math> does not depend on <math>n</math>. What is <math>S_{20}</math>?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420</math>
  
 
[[2022 AMC 10B Problems/Problem 15|Solution]]
 
[[2022 AMC 10B Problems/Problem 15|Solution]]
  
 
==Problem 16 ==
 
==Problem 16 ==
−
XXX
+
The diagram below shows a rectangle with side lengths <math>4</math> and <math>8</math> and a square with side length <math>5</math>. 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>
 +
size(5cm);
 +
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
 +
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
 +
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
 +
label("$4$", (8,2), E);
 +
label("$8$", (4,0), S);
 +
label("$5$", (3,11/2), NW);
 +
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(1.1));
 +
</asy>
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<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 10B Problems/Problem 16|Solution]]
 
[[2022 AMC 10B Problems/Problem 16|Solution]]
  
 
==Problem 17 ==
 
==Problem 17 ==
−
XXX
+
One of the following numbers is not divisible by any prime number less than <math>10.</math> Which is it?
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<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 10B Problems/Problem 17|Solution]]
 
[[2022 AMC 10B Problems/Problem 17|Solution]]
  
 
==Problem 18 ==
 
==Problem 18 ==
−
XXX
+
Consider systems of three linear equations with unknowns <math>x</math>, <math>y</math>, and <math>z</math>,
 +
<cmath>
 +
\begin{align*}
 +
a_1 x + b_1 y + c_1 z & = 0 \\
 +
a_2 x + b_2 y + c_2 z & = 0 \\
 +
a_3 x + b_3 y + c_3 z & = 0
 +
\end{align*}
 +
</cmath>
 +
where each of the coefficients is either <math>0</math> or <math>1</math> and the system has a solution other than <math>x=y=z=0</math>.
 +
For example, one such system is <cmath>\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}</cmath>
 +
with a nonzero solution of <math>\{x,y,z\} = \{1, -1, 1\}</math>. How many such systems of equations are there?
 +
(The equations in a system need not be distinct, and two systems containing the same equations in a
 +
different order are considered different.)
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 302 \qquad\textbf{(B)}\ 338 \qquad\textbf{(C)}\ 340 \qquad\textbf{(D)}\ 343 \qquad\textbf{(E)}\ 344</math>
  
 
[[2022 AMC 10B Problems/Problem 18|Solution]]
 
[[2022 AMC 10B Problems/Problem 18|Solution]]
  
 
==Problem 19 ==
 
==Problem 19 ==
−
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,mediumgray,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,mediumgray,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));
 +
        }
  
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
        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 10B Problems/Problem 19|Solution]]
 
[[2022 AMC 10B Problems/Problem 19|Solution]]
Line 159: Line 290:
  
 
==Problem 21 ==
 
==Problem 21 ==
−
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)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23</math>
  
 
[[2022 AMC 10B Problems/Problem 21|Solution]]
 
[[2022 AMC 10B Problems/Problem 21|Solution]]
Line 180: Line 311:
  
 
==Problem 24 ==
 
==Problem 24 ==
−
XXX
+
Consider functions <math>f</math> that satisfy <cmath>|f(x)-f(y)|\leq \frac{1}{2}|x-y|</cmath> for all real numbers <math>x</math> and <math>y</math>. Of all such functions that also satisfy the equation <math>f(300) = f(900)</math>, what is the greatest possible value of
−
 
+
<cmath>f(f(800))-f(f(400))?</cmath>
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
<math>\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 200</math>
  
 
[[2022 AMC 10B Problems/Problem 24|Solution]]
 
[[2022 AMC 10B Problems/Problem 24|Solution]]
  
 
==Problem 25 ==
 
==Problem 25 ==
−
XXX
+
Let <math>x_0,x_1,x_2,\dotsc</math> be a sequence of numbers, where each <math>x_k</math> is either <math>0</math> or <math>1</math>. For each positive integer <math>n</math>, define
−
 
+
<cmath>S_n = \sum_{k=0}^{n-1} x_k 2^k</cmath>
−
<math>\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X</math>
+
Suppose <math>7S_n \equiv 1 \pmod{2^n}</math> for all <math>n \geq 1</math>. What is the value of the sum 
 +
<cmath>x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?</cmath>
 +
<math>\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) }12\qquad \textbf{(D) } 14\qquad \textbf{(E) }15</math>
  
 
[[2022 AMC 10B Problems/Problem 25|Solution]]
 
[[2022 AMC 10B Problems/Problem 25|Solution]]
  
 
==See also==
 
==See also==
−
{{AMC10 box|year=2022|ab=B|before=[[2021 Fall AMC 10B Problems]]|after=[[2023 AMC 10B Problems]]}}
+
{{AMC10 box|year=2022|ab=B|before=[[2022 AMC 10A Problems]]|after=[[2023 AMC 10A Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 05:09, 31 October 2024

2022 AMC 10B (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 SAT 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 three-digit positive integers have an odd number of even digits?

$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

Solution

Problem 4

A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$th hiccup occur?

$\textbf{(A) }15 \text{ seconds after } 4:58$

$\textbf{(B) }20 \text{ seconds after } 4:58$

$\textbf{(C) }25 \text{ seconds after } 4:58$

$\textbf{(D) }30 \text{ seconds after } 4:58$

$\textbf{(E) }35 \text{ seconds after } 4:58$

Solution

Problem 5

What is the value of \[\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?\] $\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}$

Solution

Problem 6

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

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

Solution

Problem 7

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 8

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,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 9

The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$?

$\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Solution

Problem 10

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 11

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?

$\textbf{(A) }$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(B) }$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

$\textbf{(C) }$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(D) }$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

$\textbf{(E) }$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

Solution

Problem 12

A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?

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

Solution

Problem 13

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 16$

Solution

Problem 14

Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \ldots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain?

$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16$

Solution

Problem 15

Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$?

$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$

Solution

Problem 16

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] size(5cm); filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray); draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1)); draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1)); label("$4$", (8,2), E); label("$8$", (4,0), S); label("$5$", (3,11/2), NW); draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(1.1)); [/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 17

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 18

Consider systems of three linear equations with unknowns $x$, $y$, and $z$, \begin{align*} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=z=0$. For example, one such system is \[\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}\] with a nonzero solution of $\{x,y,z\} = \{1, -1, 1\}$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)

$\textbf{(A)}\ 302 \qquad\textbf{(B)}\ 338 \qquad\textbf{(C)}\ 340 \qquad\textbf{(D)}\ 343 \qquad\textbf{(E)}\ 344$

Solution

Problem 19

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,mediumgray,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,mediumgray,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 20

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$?

$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$

Solution

Problem 21

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 22

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 23

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 24

Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of \[f(f(800))-f(f(400))?\] $\textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 200$

Solution

Problem 25

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 $n \geq 1$. 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

See also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
2022 AMC 10A Problems 		Followed by
2023 AMC 10A Problems
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All AMC 10 Problems and Solutions

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