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Difference between revisions of "2023 AMC 12A Problems"

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Don't post the leaked problems here!
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{{AMC12 Problems|year=2023|ab=A}}
 +
 
 +
==Problem 1==
 +
 
 +
Cities <math>A</math> and <math>B</math> are <math>45</math> miles apart. Alicia lives in <math>A</math> and Beth lives in <math>B</math>. Alicia bikes towards <math>B</math> at 18 miles per hour. Leaving at the same time, Beth bikes toward <math>A</math> at 12 miles per hour. How many miles from City <math>A</math> will they be when they meet?
 +
 
 +
<math>\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 1|Solution]]
 +
 
 +
==Problem 2==
 +
 
 +
The weight of <math>\frac{1}{3}</math> of a large pizza together with <math>3 \frac{1}{2}</math> cups of orange slices is the same weight of <math>\frac{3}{4}</math> of a large pizza together with <math>\frac{1}{2}</math> cups of orange slices. A cup of orange slices weigh <math>\frac{1}{4}</math> of a pound. What is the weight, in pounds, of a large pizza?
 +
 
 +
<math>\textbf{(A) }1\frac{4}{5}\qquad\textbf{(B) }2\qquad\textbf{(C) }2\frac{2}{5}\qquad\textbf{(D) }3\qquad\textbf{(E) }3\frac{3}{5}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
 
 +
How many positive perfect squares less than <math>2023</math> are divisible by <math>5</math>?
 +
 
 +
<math>\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
 
 +
How many digits are in the base-ten representation of <math>8^5 \cdot 5^{10} \cdot 15^5</math>?
 +
 
 +
<math>\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 4|Solution]]
 +
 
 +
==Problem 5==
 +
 
 +
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
 +
 
 +
<math>\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
 
 +
Points <math>A</math> and <math>B</math> lie on the graph of <math>y=\log_{2}x</math>. The midpoint of <math>\overline{AB}</math> is <math>(6, 2)</math>. What is the positive difference between the <math>x</math>-coordinates of <math>A</math> and <math>B</math>?
 +
 
 +
<math>\textbf{(A)}~2\sqrt{11}\qquad\textbf{(B)}~4\sqrt{3}\qquad\textbf{(C)}~8\qquad\textbf{(D)}~4\sqrt{5}\qquad\textbf{(E)}~9</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
 
 +
A digital display shows the current date as an <math>8</math>-digit integer consisting of a <math>4</math>-digit year, followed by a <math>2</math>-digit month, followed by a <math>2</math>-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in <math>2023</math> will each digit appear an even number of times in the 8-digital display for that date?
 +
 
 +
<math>\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
 
 +
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an <math>11</math> on the next quiz, her mean will increase by <math>1</math>. If she scores an <math>11</math> on each of the next three quizzes, her mean will increase by <math>2</math>. What is the mean of her quiz scores currently?
 +
 
 +
<math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
 
 +
A square of area <math>2</math> is inscribed in a square of area <math>3</math>, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
 +
<asy>
 +
size(200);
 +
defaultpen(linewidth(0.6pt)+fontsize(10pt));
 +
real y = sqrt(3);
 +
pair A,B,C,D,E,F,G,H;
 +
A = (0,0);
 +
B = (0,y);
 +
C = (y,y);
 +
D = (y,0);
 +
E = ((y + 1)/2,y);
 +
F = (y, (y - 1)/2);
 +
G = ((y - 1)/2, 0);
 +
H = (0,(y + 1)/2);
 +
fill(H--B--E--cycle, gray);
 +
draw(A--B--C--D--cycle);
 +
draw(E--F--G--H--cycle);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
 
 +
Positive real numbers <math>x</math> and <math>y</math> satisfy <math>y^3 = x^2</math> and <math>(y-x)^2 = 4y^2</math>. What is <math>x+y</math>?
 +
 
 +
<math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
 
 +
What is the degree measure of the acute angle formed by lines with slopes <math>2</math> and <math>\tfrac{1}{3}</math>?
 +
 
 +
<math>\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
 
 +
What is the value of
 +
<cmath> 2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?</cmath>
 +
 
 +
<math>\textbf{(A) } 2023 \qquad\textbf{(B) } 2679 \qquad\textbf{(C) } 2941 \qquad\textbf{(D) } 3159 \qquad\textbf{(E) } 3235</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
 
 +
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was <math>40\%</math> more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
 +
 
 +
<math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
 
 +
How many complex numbers satisfy the equation <math>z^{5}=\overline{z}</math>, where <math>\overline{z}</math> is the conjugate of the complex number <math>z</math>?
 +
 
 +
<math>\textbf{(A)}~2\qquad\textbf{(B)}~3\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
 
 +
Usain is walking for exercise by zigzagging across a <math>100</math>-meter by <math>30</math>-meter rectangular field, beginning at point <math>A</math> and ending on the segment <math>\overline{BC}</math>. He wants to increase the distance walked by zigzagging as shown in the figure below <math>(APQRS)</math>. What angle <math>\theta</math><math>\angle PAB=\angle QPC=\angle RQB=\cdots</math> will produce in a length that is <math>120</math> meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)
 +
 
 +
<math>\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
 
 +
Consider the set of complex numbers <math>z</math> satisfying <math>|1+z+z^{2}|=4</math>. The maximum value of the imaginary part of <math>z</math> can be written in the form <math>\tfrac{\sqrt{m}}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>?
 +
 
 +
<math>\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\qquad\textbf{(E)}~24</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
 
 +
Flora the frog starts at <math>0</math> on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance <math>m</math> with probability <math>\frac{1}{2^m}</math>. What is the probability that Flora will eventually land at <math>10</math>?
 +
 
 +
<math>\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
 
 +
Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
 +
 
 +
<asy>
 +
import olympiad;
 +
size(10cm);
 +
draw(circle((0,0),0.75));
 +
draw(circle((-0.25,0),1));
 +
draw(circle((0.25,0),1));
 +
draw(circle((0,6/7),3/28));
 +
pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
 +
dot(B^^C);
 +
draw(B--E, dashed);
 +
draw(C--F, dashed);
 +
draw(B--C);
 +
label("$C_4$", D);
 +
label("$C_1$", (-1.375, 0));
 +
label("$C_2$", (1.375,0));
 +
label("$\frac{1}{2}$", (0, -.125));
 +
label("$C_3$", (-0.4, -0.4));
 +
label("$1$", (-.85, 0.70));
 +
label("$1$", (.85, -.7));
 +
import olympiad;
 +
markscalefactor=0.005;
 +
</asy>
 +
 
 +
<math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
 
 +
What is the product of all the solutions to the equation<cmath>\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?</cmath>
 +
<math>\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
 
 +
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
 +
 
 +
<asy>
 +
size(4.5cm);
 +
label("$1$", (0,0));
 +
label("$1$", (-0.5,-2/3));
 +
label("$1$", (0.5,-2/3));
 +
label("$1$", (-1,-4/3));
 +
label("$3$", (0,-4/3));
 +
label("$1$", (1,-4/3));
 +
label("$1$", (-1.5,-2));
 +
label("$5$", (-0.5,-2));
 +
label("$5$", (0.5,-2));
 +
label("$1$", (1.5,-2));
 +
label("$1$", (-2,-8/3));
 +
label("$7$", (-1,-8/3));
 +
label("$11$", (0,-8/3));
 +
label("$7$", (1,-8/3));
 +
label("$1$", (2,-8/3));
 +
</asy>
 +
 
 +
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?
 +
 
 +
<math>\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
 
 +
If <math>A</math> and <math>B</math> are vertices of a polyhedron, define the distance <math>d(A, B)</math> to be the minimum number of edges of the polyhedron one must traverse in order to connect <math>A</math> and <math>B</math>. For example, if <math>\overline{AB}</math> is an edge of the polyhedron, then <math>d(A, B) = 1</math>, but if <math>\overline{AC}</math> and <math>\overline{CB}</math> are edges and <math>\overline{AB}</math> is not an edge, then <math>d(A, B) = 2</math>. Let <math>Q</math>, <math>R</math>, and <math>S</math> be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that <math>d(Q, R) > d(R, S)</math>?
 +
 
 +
<math>\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
 
 +
Let <math>f</math> be the unique function defined on the positive integers such that<cmath>\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1</cmath>for all positive integers <math>n</math>, where the sum is taken over all positive divisors of <math>n</math>. What is <math>f(2023)</math>?
 +
 
 +
<math>\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
 
 +
How many ordered pairs of positive real numbers <math>(a,b)</math> satisfy the equation
 +
<cmath>(1+2a)(2+2b)(2a+b) = 32ab?</cmath>
 +
 
 +
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
 
 +
Let <math>K</math> be the number of sequences <math>A_1</math>, <math>A_2</math>, <math>\dots</math>, <math>A_n</math> such that <math>n</math> is a positive integer less than or equal to <math>10</math>, each <math>A_i</math> is a subset of <math>\{1, 2, 3, \dots, 10\}</math>, and <math>A_{i-1}</math> is a subset of <math>A_i</math> for each <math>i</math> between <math>2</math> and <math>n</math>, inclusive. For example, <math>\{\}</math>, <math>\{5, 7\}</math>, <math>\{2, 5, 7\}</math>, <math>\{2, 5, 7\}</math>, <math>\{2, 5, 6, 7, 9\}</math> is one such sequence, with <math>n = 5</math>.What is the remainder when <math>K</math> is divided by <math>10</math>?
 +
 
 +
<math>\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
 
 +
There is a unique sequence of integers <math>a_1, a_2, \cdots a_{2023}</math> such that
 +
<cmath>
 +
\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}
 +
</cmath>whenever <math>\tan 2023x</math> is defined. What is <math>a_{2023}?</math>
 +
 
 +
<math>\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023</math>
 +
 
 +
[[2023 AMC 12A Problems/Problem 25|Solution]]
 +
 
 +
==See also==
 +
{{AMC12 box|year=2023|ab=A|before=[[2022 AMC 12B Problems]]|after=[[2023 AMC 12B Problems]]}}
 +
* [[AMC 12]]
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competitions]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Revision as of 23:12, 9 November 2023

2023 AMC 12A (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

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet?

$\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Solution

Problem 2

The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cups of orange slices. A cup of orange slices weigh $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza?

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

Solution

Problem 3

How many positive perfect squares less than $2023$ are divisible by $5$?

$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$

Solution

Problem 4

How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?

$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$

Solution

Problem 5

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?

$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

Solution

Problem 6

Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$?

$\textbf{(A)}~2\sqrt{11}\qquad\textbf{(B)}~4\sqrt{3}\qquad\textbf{(C)}~8\qquad\textbf{(D)}~4\sqrt{5}\qquad\textbf{(E)}~9$

Solution

Problem 7

A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?

$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$

Solution

Problem 8

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Solution

Problem 9

A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? [asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]

$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$

Solution

Problem 10

Positive real numbers $x$ and $y$ satisfy $y^3 = x^2$ and $(y-x)^2 = 4y^2$. What is $x+y$?

$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 42$

Solution

Problem 11

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$?

$\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$

Solution

Problem 12

What is the value of \[2^3 - 1^2 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]

$\textbf{(A) } 2023 \qquad\textbf{(B) } 2679 \qquad\textbf{(C) } 2941 \qquad\textbf{(D) } 3159 \qquad\textbf{(E) } 3235$

Solution

Problem 13

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$

Solution

Problem 14

How many complex numbers satisfy the equation $z^{5}=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$?

$\textbf{(A)}~2\qquad\textbf{(B)}~3\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

Solution

Problem 15

Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta$$\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)

$\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}$

Solution

Problem 16

Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

$\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\qquad\textbf{(E)}~24$

Solution

Problem 17

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$?

$\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$

Solution

Problem 18

Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?

[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label("$C_4$", D); label("$C_1$", (-1.375, 0)); label("$C_2$", (1.375,0)); label("$\frac{1}{2}$", (0, -.125)); label("$C_3$", (-0.4, -0.4)); label("$1$", (-.85, 0.70)); label("$1$", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]

$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

Solution

Problem 19

What is the product of all the solutions to the equation\[\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?\] $\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$

Solution

Problem 20

Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:

[asy] size(4.5cm); label("$1$", (0,0)); label("$1$", (-0.5,-2/3)); label("$1$", (0.5,-2/3)); label("$1$", (-1,-4/3)); label("$3$", (0,-4/3)); label("$1$", (1,-4/3)); label("$1$", (-1.5,-2)); label("$5$", (-0.5,-2)); label("$5$", (0.5,-2)); label("$1$", (1.5,-2)); label("$1$", (-2,-8/3)); label("$7$", (-1,-8/3)); label("$11$", (0,-8/3)); label("$7$", (1,-8/3)); label("$1$", (2,-8/3)); [/asy]

Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?

$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$

Solution

Problem 21

If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?

$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$

Solution

Problem 22

Let $f$ be the unique function defined on the positive integers such that\[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\]for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$?

$\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$

Solution

Problem 23

How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\]

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}$

Solution

Problem 24

Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$.What is the remainder when $K$ is divided by $10$?

$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

Solution

Problem 25

There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$

$\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023$

Solution

See also

2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by
2022 AMC 12B Problems 		Followed by
2023 AMC 12B Problems
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All AMC 12 Problems and Solutions

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