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Difference between revisions of "2022 AMC 12A Problems"

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==Problem 6==
 
==Problem 6==
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The sum of three numbers is <math>96.</math> The first number is <math>6</math> times the third number, and the third number is <math>40</math> less than the second number. What is the absolute value of the difference between the first and second numbers?
+
A data set consists of <math>6</math> not distinct) positive integers: <math>1</math>, <math>7</math>, <math>5</math>, <math>2</math>, <math>5</math>, and <math>X</math>. The
 +
average (arithmetic mean) of the <math>6</math> numbers equals a value in the data set. What is
 +
the sum of all positive values of <math>X</math>?
  
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<math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5</math>
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<math>\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40</math>
  
 
[[2022 AMC 12A Problems/Problem 6|Solution]]
 
[[2022 AMC 12A Problems/Problem 6|Solution]]

Revision as of 00:24, 12 November 2022

2022 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

What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac13}}?\] $\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}$

Solution

Problem 2

The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?

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

Solution

Problem 3

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 4

The least common multiple of a positive divisor $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?

$\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12$

Solution

Problem 5

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 6

A data set consists of $6$ not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?

$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

Solution

Problem 7

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]

$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$

Solution

Problem 8

The infinite product

$\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} ...$

evaluates to a real number. What is that number?


Solution

Problem 9

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 10

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 11

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 12

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 13

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 14

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 15

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 16

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 17

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 18

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 19

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 20

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 21

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 22

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 23

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 24

These problems will be posted once the 2022 AMC 12A is released.

Solution

Problem 25

These problems will be posted once the 2022 AMC 12A is released.

Solution

See also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
2021 AMC 12B Problems 		Followed by
2022 AMC 12B Problems
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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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