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Difference between revisions of "2023 AMC 8 Problems"

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==Problem 21==
 
==Problem 21==
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Alina writes the numbers <math>1, 2, \dots , 9</math> on separate cards, one number per card. She wishes to divide the cards into <math>3</math> groups of <math>3</math> cards so that the sum of the number in each group will be the same. In how many ways can this be done?
  
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<math>\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X</math>
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<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
  
 
[[2023 AMC 8 Problems/Problem 21|Solution]]
 
[[2023 AMC 8 Problems/Problem 21|Solution]]

Revision as of 18:46, 24 January 2023

2023 AMC 8 (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 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 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

TODO: transcribe from [1]

Problem 1

What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Solution

Problem 2

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 3

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 4

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 5

A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?

$\textbf{(A)}\ 1250 \qquad \textbf{(B)}\ 1500 \qquad \textbf{(C)}\ 1750 \qquad \textbf{(D)}\ 1800 \qquad \textbf{(E)}\ 2000$

Solution

Problem 6

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 7

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 8

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 9

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 10

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac[1}{3}$ (Error compiling LaTeX. Unknown error_msg) of what the moose left behind. How much of the original pie still remained after the porcupine left?

$\textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{5}{12}$

Solution

Problem 11

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 12

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 13

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 14

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 15

Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time?

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 4.8 \qquad \textbf{(E)}\ 5$

Solution

Problem 16

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 17

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 18

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 19

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 20

Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of two additional numbers?

$\textbf{(A)}\ 56 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 58 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 61$

Solution

Problem 21

Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done?

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

Solution

Problem 22

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10$

Solution

Problem 23

Each square in a 3x3 grid is randomly filled with one of the 4 gray-and-white tiles as shown below.

[insert asy]

What is the probability that the tiling will contain a large gray diamond in one of the smaller 2x2 grids? Below is such a tiling

[insert asy]

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 24

$\textbf{(A)}\ X \qquad \textbf{(B)}\ X \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ X \qquad \textbf{(E)}\ X$

Solution

Problem 25

Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that \[1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace 241 \le a_{15}\le 250.\] What is the sum of digits of $a_{14}$?

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

Solution

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
2022 AMC 8 		Followed by
2024 AMC 8
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 AJHSME/AMC 8 Problems and Solutions
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