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Difference between revisions of "2021 Mock AMC 8 Problems"

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\qquad\mathrm{(D)}\ 7
 
\qquad\mathrm{(D)}\ 7
 
\qquad\mathrm{(E)}\ 25</math>
 
\qquad\mathrm{(E)}\ 25</math>
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https://artofproblemsolving.com/wiki/index.php/2021_Mock_AMC_8_Problems/Problem_1
  
 
==Problem 2==
 
==Problem 2==
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<math>\mathrm{(A)}\ 4
 
<math>\mathrm{(A)}\ 4
−
\qquad\mathrm{(B)}\ 12
+
\qquad\mathrm{(B)}\ 16
−
\qquad\mathrm{(C)}\ 24
+
\qquad\mathrm{(C)}\ 256
−
\qquad\mathrm{(D)}\ 72
+
\qquad\mathrm{(D)}\ 288
−
\qquad\mathrm{(E)}\ 144</math>
+
\qquad\mathrm{(E)}\ 576</math>
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 +
https://artofproblemsolving.com/wiki/index.php/2021_Mock_AMC_8_Problems/Problem_2
  
 
==Problem 3==
 
==Problem 3==
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==Problem 4==
 
==Problem 4==
−
A rectangle with nonnegative integer side lengths has area <math>2021</math>. In how many ways is this possible?
+
A rectangle with positive integer side lengths has area <math>2021</math>. In how many ways is this possible?
  
 
<math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5</math>
 
<math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5</math>
 +
 +
==Problem 5==
 +
Fiona leaves her house to go to the airport. She drives for <math>30</math> minutes at a constant rate of <math>40</math> miles per hour, then walks for <math>20</math> minutes at a constant rate of <math>3.6</math> miles per hour. She then goes on a train going 50 miles per hour for 12 minutes. How far has she traveled?
 +
 +
<math>\textbf{(A) } 31.2 \qquad \textbf{(B) } 32.2 \qquad \textbf{(C) } 32.4 \qquad \textbf{(D) } 33.8 \qquad \textbf{(E) } 35.9</math>
 +
 +
==Problem 6==
 +
Hexagon <math>ABCDEF</math> has side length <math>4</math>. What is the area of this hexagon rounded to the nearest tenth?
 +
 +
<math>\textbf{(A) } 40.8 \qquad \textbf{(B) } 41.5 \qquad \textbf{(C) } 41.6 \qquad \textbf{(D) } 42.4 \qquad \textbf{(E) } 44.3</math>
 +
 +
==Problem 7==
 +
The number <math>N</math> is a positive <math>3</math> digit integer.
 +
 +
•When <math>N</math> is divided by <math>80</math>, the remainder is <math>4</math>
 +
 +
•When <math>N</math> is divided by <math>3</math>, the remainder is <math>1</math>
 +
 +
•<math>N</math> is a perfect square.
 +
 +
What is the sum of the digits of <math>N</math>?
 +
 +
<math>\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad \textbf{(E) }16</math>
 +
 +
==Problem 8==
 +
How many many zeros are at the right of the last nonzero digit of the number <math>1020!</math>?
 +
 +
<math>\textbf{(A) }204\qquad\textbf{(B) }244\qquad\textbf{(C) }252\qquad\textbf{(D) }253\qquad \textbf{(E) }254</math>
 +
 +
==Problem 9==
 +
Isosceles trapezoid <math>ABCD</math> has <math>AB = 8</math>. Point <math>E</math> is on <math>DC</math> such that <math>AE</math> is perpendicular to <math>DC</math> and that <math>AE</math> = <math>9</math>. <math>BC</math> and <math>AE</math> are extended to point <math>F</math> to make isosceles triangle <math>FCD</math>. Point <math>F</math> is <math>18</math> units away from the midpoint of <math>AB</math>. What is the area of isosceles trapezoid <math>ABCD</math>?
 +
 +
<math>\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 90 \qquad \text{(D)}\ 108 \qquad \text{(E)}\ 162</math>
 +
 +
==Problem 10==
 +
Maddie picks <math>2</math> numbers between <math>0</math> and <math>1</math>. The probability that both numbers are less than <math>\frac {2}{3}</math> can be expressed in the form <math>\frac {a}{b}</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a</math> + <math>b</math>?
 +
 +
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 18</math>
 +
 +
==Problem 11==
 +
Which of the following numbers is the smallest?
 +
 +
<math>3 \sqrt 2</math>, <math>\frac {21}{5}</math>, <math>\frac {5}{2} \sqrt 3</math>, <math>2 \sqrt 5</math>, <math>\frac {5}{3} \sqrt 7</math>
 +
 +
<math> \textbf{(A)}\ 3 \sqrt 2 \qquad\textbf{(B)}\ \frac {21}{5} \qquad\textbf{(C)}\ \frac {5}{2} \sqrt 3 \qquad\textbf{(D)}\ 2 \sqrt 5 \qquad\textbf{(E)}\ \frac {5}{3} \sqrt 7 </math>
 +
 +
==Problem 12==
 +
Isosceles <math>\triangle ABC</math> has <math>\angle ABC</math> = <math>80</math> degrees. What is the sum of all possible values for <math>\angle CAB</math>?
 +
 +
<math>\textbf{(A)} ~80 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~130 \qquad\textbf{(D)} ~140 \qquad\textbf{(E)} ~150</math>
 +
 +
==Problem 13==
 +
Real numbers <math>x</math> and <math>y</math> has these following conditions:
 +
 +
<math>x \cdot y</math> = <math>43</math>
 +
 +
<math>x + y = 37</math>
 +
 +
What is the product of the roots of the equation?

Latest revision as of 16:33, 9 November 2023

Problem 1

What is the value of $1-2+3-4+5-6+7-8+9$?

$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7 \qquad\mathrm{(E)}\ 25$

https://artofproblemsolving.com/wiki/index.php/2021_Mock_AMC_8_Problems/Problem_1

Problem 2

Aaron has a rectangular yard measuring $4$ feet by $10$ feet. How many $2$ inch by $5$ inch rectangular bricks can he fit in his yard?

$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 256 \qquad\mathrm{(D)}\ 288 \qquad\mathrm{(E)}\ 576$

https://artofproblemsolving.com/wiki/index.php/2021_Mock_AMC_8_Problems/Problem_2

Problem 3

Amy, Bob, Cassie, and Darren are on a camping trip. Each of them has $4$ choices for what they wear on day $1$ of the camping trip. How many different arrangements of what they wear are possible on day $1$ of the camping trip?

$\textbf{(A) } 16 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 64 \qquad \textbf{(D) } 256 \qquad \textbf{(E) 500}$

Problem 4

A rectangle with positive integer side lengths has area $2021$. In how many ways is this possible?

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

Problem 5

Fiona leaves her house to go to the airport. She drives for $30$ minutes at a constant rate of $40$ miles per hour, then walks for $20$ minutes at a constant rate of $3.6$ miles per hour. She then goes on a train going 50 miles per hour for 12 minutes. How far has she traveled?

$\textbf{(A) } 31.2 \qquad \textbf{(B) } 32.2 \qquad \textbf{(C) } 32.4 \qquad \textbf{(D) } 33.8 \qquad \textbf{(E) } 35.9$

Problem 6

Hexagon $ABCDEF$ has side length $4$. What is the area of this hexagon rounded to the nearest tenth?

$\textbf{(A) } 40.8 \qquad \textbf{(B) } 41.5 \qquad \textbf{(C) } 41.6 \qquad \textbf{(D) } 42.4 \qquad \textbf{(E) } 44.3$

Problem 7

The number $N$ is a positive $3$ digit integer.

•When $N$ is divided by $80$, the remainder is $4$

•When $N$ is divided by $3$, the remainder is $1$

•$N$ is a perfect square.

What is the sum of the digits of $N$?

$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad \textbf{(E) }16$

Problem 8

How many many zeros are at the right of the last nonzero digit of the number $1020!$?

$\textbf{(A) }204\qquad\textbf{(B) }244\qquad\textbf{(C) }252\qquad\textbf{(D) }253\qquad \textbf{(E) }254$

Problem 9

Isosceles trapezoid $ABCD$ has $AB = 8$. Point $E$ is on $DC$ such that $AE$ is perpendicular to $DC$ and that $AE$ = $9$. $BC$ and $AE$ are extended to point $F$ to make isosceles triangle $FCD$. Point $F$ is $18$ units away from the midpoint of $AB$. What is the area of isosceles trapezoid $ABCD$?

$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 90 \qquad \text{(D)}\ 108 \qquad \text{(E)}\ 162$

Problem 10

Maddie picks $2$ numbers between $0$ and $1$. The probability that both numbers are less than $\frac {2}{3}$ can be expressed in the form $\frac {a}{b}$ where $a$ and $b$ are relatively prime positive integers. What is $a$ + $b$?

$\text{(A)}\ 5 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 18$

Problem 11

Which of the following numbers is the smallest?

$3 \sqrt 2$, $\frac {21}{5}$, $\frac {5}{2} \sqrt 3$, $2 \sqrt 5$, $\frac {5}{3} \sqrt 7$

$\textbf{(A)}\ 3 \sqrt 2 \qquad\textbf{(B)}\ \frac {21}{5} \qquad\textbf{(C)}\ \frac {5}{2} \sqrt 3 \qquad\textbf{(D)}\ 2 \sqrt 5 \qquad\textbf{(E)}\ \frac {5}{3} \sqrt 7$

Problem 12

Isosceles $\triangle ABC$ has $\angle ABC$ = $80$ degrees. What is the sum of all possible values for $\angle CAB$?

$\textbf{(A)} ~80 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~130 \qquad\textbf{(D)} ~140 \qquad\textbf{(E)} ~150$

Problem 13

Real numbers $x$ and $y$ has these following conditions:

$x \cdot y$ = $43$

$x + y = 37$

What is the product of the roots of the equation?

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