Difference between revisions of "2022 AMC 8 Problems"

Revision as of 12:06, 28 January 2022

IMPORTANT: When copied a problem, replace the X's for answer choices.

Problem 1

The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

usepackage("mathptmx");
defaultpen(linewidth(0.5));
size(5cm);
defaultpen(fontsize(14pt));
label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7));
label("$\textbf{Team}$", (2.1,3)--(3.9,3));
filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5);

draw((0,0)--(6,0), gray);
draw((0,1)--(6,1), gray);
draw((0,2)--(6,2), gray);
draw((0,3)--(6,3), gray);
draw((0,4)--(6,4), gray);
draw((0,5)--(6,5), gray);
draw((0,6)--(6,6), gray);

draw((0,0)--(0,6), gray);
draw((1,0)--(1,6), gray);
draw((2,0)--(2,6), gray);
draw((3,0)--(3,6), gray);
draw((4,0)--(4,6), gray);
draw((5,0)--(5,6), gray);
draw((6,0)--(6,6), gray);
 (Error making remote request. Unexpected URL sent back)

$\textbf{(A) } 10 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

Solution

Problem 2

Consider these two operations: $\begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*}$ What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$

$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$

Solution

Problem 3

When three positive integers $a$ , $b$ , and $c$ are multiplied together, their product is $100$ . Suppose $a < b < c$ . In how many ways can the numbers be chosen?

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

Solution

Problem 4

The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$ . What is the resulting image?

$[asy] // pog diagram usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5); [/asy]$

$[asy] // pog diagram usepackage("newtxtext"); size(12.5cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$\textbf{(A)}$",(-1,1),W); draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7)); draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); label("$\textbf{(B)}$",(2,1),W); draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2)); draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); label("$\textbf{(C)}$",(5,1),W); draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1)); draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); label("$\textbf{(D)}$",(-1,-1.5),W); draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75)); draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); label("$\textbf{(E)}$",(2,-1.5),W); [/asy]$

Solution

Problem 5

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?

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

Solution

Problem 6

Three positive integers are equally spaced on a number line. The middle number is $15,$ and the largest number is $4$ times the smallest number. What is the smallest of these three numbers?

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

Solution

Problem 7

When the World Wide Web first became popular in the $1990$ s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$ -megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.)

$\textbf{(A) } 0.6 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 1800 \qquad \textbf{(D) } 7200 \qquad \textbf{(E) } 36000$

Solution

Problem 8

What is the value of $\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?$

$\textbf{(A) } \frac{1}{462} \qquad \textbf{(B) } \frac{1}{231} \qquad \textbf{(C) } \frac{1}{132} \qquad \textbf{(D) } \frac{2}{213} \qquad \textbf{(E) } \frac{1}{22}$

Solution

Problem 9

A cup of boiling water ( $212^{\circ}\text{F}$ ) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$ . Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?

$\textbf{(A)} ~77\qquad\textbf{(B)} ~86\qquad\textbf{(C)} ~92\qquad\textbf{(D)} ~98\qquad\textbf{(E)} ~104$

Solution

Problem 10

One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$ , drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$ . After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?

$[asy] unitsize(12); usepackage("mathptmx"); defaultpen(fontsize(8)+linewidth(.7)); int mod12(int i) {if (i<13) {return i;} else {return i-12;}} void drawgraph(pair sh,string lab) { for (int i=0;i<11;++i) { for (int j=0;j<6;++j) { draw(shift(sh+(i,j))*unitsquare,mediumgray); } } draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8)); draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8)); for (int i=1;i<10;++i) { draw(shift(sh)*((i,-.2)--(i,.2))); } label("8\tiny{\textsc{am}}",sh+(1,-.2),S); for (int i=2;i<9;++i) { label(string(mod12(i+7)),sh+(i,-.2),S); } label("4\tiny{\textsc{pm}}",sh+(9,-.2),S); for (int i=1;i<6;++i) { label(string(30*i),sh+(0,i),2*W); } draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10)); label("$\textbf{("+lab+")}$",sh+(-2.1,6.8),fontsize(10)); } drawgraph((0,0),"A"); drawgraph((15,0),"B"); drawgraph((0,-10),"C"); drawgraph((15,-10),"D"); drawgraph((0,-20),"E"); dotfactor=6; draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9)); dot((1,0)^^(3,3)^^(6,3)^^(8,0)); pair sh = (15,0); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0)); pair sh = (0,-10); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0)); pair sh = (15,-10); draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0)); pair sh = (0,-20); draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0)); [/asy]$

Solution

Problem 11

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with?

$\textbf{(A)} ~34\qquad\textbf{(B)} ~38\qquad\textbf{(C)} ~41\qquad\textbf{(D)} ~44\qquad\textbf{(E)} ~47$

Solution

Problem 12

How many positive integers can fill the blank in the sentence below?

“One positive integer is _____ more than twice another, and the sum of the two numbers is $28$ .”

$\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10$

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

In how many ways can the letters in $BEEKEEPER$ be rearranged so that two or more $E$ s do not appear together?

$\textbf{(A) } 1 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 120$

Solution

Problem 15

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

Solution

Problem 16

Four numbers are written in a row. The average of the first two is $21,$ the average of the middle two is $26,$ and the average of the last two is $30.$ What is the average of the first and last of the numbers?

$\textbf{(A) } 24 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 26 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 28$

Solution

Problem 17

If $n$ is an even positive integer, the double factorial notation $n!!$ represents the product of all the even integers from $2$ to $n$ . For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$ . What is the units digit of the following sum? $2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$

$\textbf{(A)} ~0\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8$

Solution

Problem 18

The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle?

$\textbf{(A) } 20 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 40 \qquad \textbf{(D) } 50 \qquad \textbf{(E) } 80$

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

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

Solution

Problem 21

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

Solution

Problem 22

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

Solution

Problem 23

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

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

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

Solution

@@ Line 197: / Line 197: @@
 ==Problem 12==
-<math>\textbf{(A) } X \qquad \textbf{(B) } X \qquad \textbf{(C) } X \qquad \textbf{(D) } X \qquad \textbf{(E) } X</math>
+How many positive integers can fill the blank in the sentence below?
+“One positive integer is _____ more than twice another, and the sum of the two numbers is <math>28</math>.”
+<math>\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10</math>
 [[2022 AMC 8 Problems/Problem 12|Solution]]

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Difference between revisions of "2022 AMC 8 Problems"

Revision as of 12:06, 28 January 2022

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25