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Difference between revisions of "2023 AMC 12B Problems"

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{{AMC12 Problems|year=2023|ab=B}}
  
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==Problem 1==
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A mom is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only fills one third of the glass for the last one. How much does she need to pour from the 3 full glasses to fill all of the glasses to an equal amount?
 +
 +
<math>\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 1|Solution]]
 +
 +
==Problem 2==
 +
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
 +
 +
<math>\textbf{(A) }46\qquad\textbf{(B) }50\qquad\textbf{(C) }48\qquad\textbf{(D) }47\qquad\textbf{(E) }49</math>
 +
 +
[[2023 AMC 12B Problems/Problem 2|Solution]]
 +
 +
==Problem 3==
 +
A <math>3-4-5</math> right triangle is inscribed in circle <math>A</math>, and a <math>5-12-13</math> right triangle is inscribed in circle <math>B</math>. What is the ratio of the area of circle <math>A</math> to the area of circle <math>B</math>?
 +
 +
<math>\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 3|Solution]]
 +
 +
==Problem 4==
 +
Jackson's paintbrush makes a narrow strip that is <math>6.5</math> mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in <math>\text{cm}^2</math>?
 +
 +
<math>\textbf{(A) }162,500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1,625\qquad\textbf{(D) }1,625,000\qquad\textbf{(E) }16,250</math>
 +
 +
[[2023 AMC 12B Problems/Problem 4|Solution]]
 +
 +
==Problem 5==
 +
You are playing a game. A <math>2 \times 1</math> rectangle covers two adjacent squares (oriented either horizontally or vertically) of a <math>3 \times 3</math> grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?
 +
 +
<math>\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6</math>
 +
 +
[[2023 AMC 12B Problems/Problem 5|Solution]]
 +
 +
==Problem 6==
 +
When the roots of the polynomial
 +
 +
<math>P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}</math>
 +
 +
are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is <math>P(x)</math> positive?
 +
 +
<math>\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7</math>
 +
 +
[[2023 AMC 12B Problems/Problem 6|Solution]]
 +
 +
==Problem 7==
 +
For how many integers <math>n</math> does the expression<cmath>\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} </cmath>represent a real number, where log denotes the base <math>10</math> logarithm?
 +
 +
<math>\textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2  \qquad \textbf{(E) }901</math>
 +
 +
[[2023 AMC 12B Problems/Problem 7|Solution]]
 +
 +
==Problem 8==
 +
How many nonempty subsets <math>B</math> of <math>\{0, 1, 2, 3, \dots, 12\}</math> have the property that the number of elements in <math>B</math> is equal to the least element of <math>B</math>? For example, <math>B = \{4, 6, 8, 11\}</math> satisfies the condition.
 +
 +
<math>\textbf{(A)}\ 256 \qquad\textbf{(B)}\ 136 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 156</math>
 +
 +
[[2023 AMC 12B Problems/Problem 8|Solution]]
 +
 +
==Problem 9==
 +
What is the area of the region in the coordinate plane defined by the inequality<cmath>\left||x|-1\right|+\left||y|-1\right|\leq 1?</cmath>
 +
 +
<math>\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15</math>
 +
 +
[[2023 AMC 12B Problems/Problem 9|Solution]]
 +
 +
==Problem 10==
 +
In the <math>xy</math>-plane, a circle of radius <math>4</math> with center on the positive <math>x</math>-axis is tangent to the <math>y</math>-axis at the origin, and a circle with radius <math>10</math> with center on the positive <math>y</math>-axis is tangent to the <math>x</math>-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
 +
 +
<math>\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7}  \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}}  \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}}  \qquad\textbf{(E)}\ \dfrac{2}{5}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 10|Solution]]
 +
 +
==Problem 11==
 +
What is the maximum area of an isosceles trapezoid that has legs of length <math>1</math> and one base twice as long as the other?
 +
<math>\textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32  \qquad \textbf{(E) } \frac{3\sqrt3}4</math>
 +
 +
[[2023 AMC 12B Problems/Problem 11|Solution]]
 +
 +
==Problem 12==
 +
For complex numbers <math>u=a+bi</math> and <math>v=c+di</math>, define the binary operation <math>\otimes</math> by<cmath>u\otimes v=ac+bdi.</cmath>Suppose <math>z</math> is a complex number such that <math>z\otimes z=z^{2}+40</math>. What is <math>|z|</math>?
 +
 +
<math>\textbf{(A)}~\sqrt{10}\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~2\sqrt{6}\qquad\textbf{(D)}~6\qquad\textbf{(E)}~5\sqrt{2}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 12|Solution]]
 +
 +
==Problem 13==
 +
A rectangular box <math>\mathcal{P}</math> has distinct edge lengths <math>a, b,</math> and <math>c</math>. The sum of the lengths of all <math>12</math> edges of <math>\mathcal{P}</math> is <math>13</math>, the sum of the areas of all <math>6</math> faces of <math>\mathcal{P}</math> is <math>\frac{11}{2}</math>, and the volume of <math>\mathcal{P}</math> is <math>\frac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of <math>\mathcal{P}</math>?
 +
 +
<math>\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 13|Solution]]
 +
 +
==Problem 14==
 +
For how many ordered pairs <math>(a,b)</math> of integers does the polynomial <math>x^3+ax^2+bx+6</math> have <math>3</math> distinct integer roots?
 +
 +
<math>\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 4</math>
 +
 +
[[2023 AMC 12B Problems/Problem 14|Solution]]
 +
 +
==Problem 15==
 +
Suppose <math>a</math>, <math>b</math>, and <math>c</math> are positive integers such that<cmath>\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.</cmath>Which of the following statements are necessarily true?
 +
 +
I. If <math>\gcd(a,14)=1</math> or <math>\gcd(b,15)=1</math> or both, then <math>\gcd(c,210)=1</math>.
 +
 +
II. If <math>\gcd(c,210)=1</math>, then <math>\gcd(a,14)=1</math> or <math>\gcd(b,15)=1</math> or both.
 +
 +
III. <math>\gcd(c,210)=1</math> if and only if <math>\gcd(a,14)=\gcd(b,15)=1</math>.
 +
 +
<math>\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 15|Solution]]
 +
 +
==Problem 16==
 +
In Coinland, there are three types of coins, each worth <math>6,</math> <math>10,</math> and <math>15.</math> What is the sum of the digits of the maximum amount of money that is impossible to have?
 +
 +
[[2023 AMC 12B Problems/Problem 16|Solution]]
 +
 +
==Problem 17==
 +
Triangle ABC has side lengths in arithmetic progression, and the smallest side has length <math>6.</math> If the triangle has an angle of <math>120^\circ,</math> what is the area of <math>ABC</math>?
 +
 +
[[2023 AMC 12B Problems/Problem 17|Solution]]
 +
 +
==Problem 18==
 +
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
 +
 +
<math>\textbf{(A)}</math> Yolanda's quiz average for the academic year was 22 points higher than Zelda's.
 +
 +
<math>\textbf{(B)}</math> Zelda's quiz average for the academic year was higher than Yolanda's.
 +
 +
<math>\textbf{(C)}</math> Yolanda's quiz average for the academic year was 3 points higher than Zelda's.
 +
 +
<math>\textbf{(D)}</math> Zelda's quiz average for the academic year equaled Yolanda's.
 +
 +
<math>\textbf{(E)}</math> If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
 +
 +
[[2023 AMC 12B Problems/Problem 18|Solution]]
 +
 +
==Problem 19==
 +
Each of <math>2023</math> balls is placed in on of <math>3</math> bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
 +
 +
<math>\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 19|Solution]]
 +
 +
==Problem 20==
 +
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
 +
 +
[[2023 AMC 12B Problems/Problem 20|Solution]]
 +
 +
==Problem 21==
 +
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is 3√3 inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?
 +
 +
<math>\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi</math>
 +
 +
[[2023 AMC 12B Problems/Problem 21|Solution]]
 +
 +
==Problem 22==
 +
A real-valued function <math>f</math> has the property that for all real numbers <math>a</math> and <math>b,</math><cmath>f(a + b)  + f(a - b) = 2f(a) f(b).</cmath>Which one of the following cannot be the value of <math>f(1)?</math>
 +
 +
<math>\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2</math>
 +
 +
[[2023 AMC 12B Problems/Problem 22|Solution]]
 +
 +
==Problem 23==
 +
When <math>n</math> standard six-sided dice are rolled, the product of the numbers rolled can be any of <math>936</math> possible values. What is <math>n</math>?
 +
 +
<math>\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11</math>
 +
 +
[[2023 AMC 12B Problems/Problem 23|Solution]]
 +
 +
==Problem 24==
 +
Integers <math>a, b, c, d</math> satisfy the following:
 +
 +
<math>abcd=2^6\cdot 3^9\cdot 5^7</math>
 +
 +
<math>\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3</math>
 +
 +
<math>\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3</math>
 +
 +
<math>\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3</math>
 +
 +
<math>\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2</math>
 +
 +
<math>\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2</math>
 +
 +
<math>\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2</math>
 +
 +
Find <math>\text{gcd}(a,b,c,d)</math>.
 +
 +
[[2023 AMC 12B Problems/Problem 24|Solution]]
 +
 +
==Problem 25==
 +
A regular pentagon with area <math>\sqrt{5}+1</math> is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
 +
 +
<math>\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}</math>
 +
 +
[[2023 AMC 12B Problems/Problem 25|Solution]]
 +
 +
==See also==
 +
{{AMC12 box|year=2023|ab=B|before=[[2023 AMC 12A Problems]]|after=[[2024 AMC 12A Problems]]}}
 +
* [[AMC 12]]
 +
* [[AMC 12 Problems and Solutions]]
 +
* [[Mathematics competitions]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Revision as of 18:35, 15 November 2023

2023 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

A mom is pouring orange juice for her 4 kids into 4 identical glasses. She fills the first 3 full, but only fills one third of the glass for the last one. How much does she need to pour from the 3 full glasses to fill all of the glasses to an equal amount?

$\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

Solution

Problem 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

$\textbf{(A) }46\qquad\textbf{(B) }50\qquad\textbf{(C) }48\qquad\textbf{(D) }47\qquad\textbf{(E) }49$

Solution

Problem 3

A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?

$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$

Solution

Problem 4

Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$?

$\textbf{(A) }162,500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1,625\qquad\textbf{(D) }1,625,000\qquad\textbf{(E) }16,250$

Solution

Problem 5

You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?

$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$

Solution

Problem 6

When the roots of the polynomial

$P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}$

are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is $P(x)$ positive?

$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

Solution

Problem 7

For how many integers $n$ does the expression\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]represent a real number, where log denotes the base $10$ logarithm?

$\textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

Solution

Problem 8

How many nonempty subsets $B$ of $\{0, 1, 2, 3, \dots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition.

$\textbf{(A)}\ 256 \qquad\textbf{(B)}\ 136 \qquad\textbf{(C)}\ 108 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 156$

Solution

Problem 9

What is the area of the region in the coordinate plane defined by the inequality\[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]

$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$

Solution

Problem 10

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

$\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

Solution

Problem 11

What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other? $\textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4$

Solution

Problem 12

For complex numbers $u=a+bi$ and $v=c+di$, define the binary operation $\otimes$ by\[u\otimes v=ac+bdi.\]Suppose $z$ is a complex number such that $z\otimes z=z^{2}+40$. What is $|z|$?

$\textbf{(A)}~\sqrt{10}\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~2\sqrt{6}\qquad\textbf{(D)}~6\qquad\textbf{(E)}~5\sqrt{2}$

Solution

Problem 13

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?

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

Solution

Problem 14

For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?

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

Solution

Problem 15

Suppose $a$, $b$, and $c$ are positive integers such that\[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\]Which of the following statements are necessarily true?

I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.

II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.

III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.

$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

Solution

Problem 16

In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have?

Solution

Problem 17

Triangle ABC has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?

Solution

Problem 18

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?

$\textbf{(A)}$ Yolanda's quiz average for the academic year was 22 points higher than Zelda's.

$\textbf{(B)}$ Zelda's quiz average for the academic year was higher than Yolanda's.

$\textbf{(C)}$ Yolanda's quiz average for the academic year was 3 points higher than Zelda's.

$\textbf{(D)}$ Zelda's quiz average for the academic year equaled Yolanda's.

$\textbf{(E)}$ If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.

Solution

Problem 19

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

Solution

Problem 20

Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?

Solution

Problem 21

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is 3√3 inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

$\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi$

Solution

Problem 22

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$\[f(a + b) + f(a - b) = 2f(a) f(b).\]Which one of the following cannot be the value of $f(1)?$

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

Solution

Problem 23

When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?

$\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$

Solution

Problem 24

Integers $a, b, c, d$ satisfy the following:

$abcd=2^6\cdot 3^9\cdot 5^7$

$\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$

$\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$

$\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$

$\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$

$\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$

$\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$

Find $\text{gcd}(a,b,c,d)$.

Solution

Problem 25

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

Solution

See also

2023 AMC 12B (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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