Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 8 Problems/Problem 1"

(Created page with "--Problem-- 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? [asy]...")
 
(→‎Solution 6 (Quick))
 
(59 intermediate revisions by 21 users not shown)
Line 1: Line 1:
−
--Problem--
+
==Problem==
  
 
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?
 
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?
  
−
[asy]
+
<asy>
 +
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);
 +
</asy>
 +
 
 +
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15</math>
 +
 
 +
==Solution 1==
 +
Draw the following four lines as shown:
 +
<asy>
 
usepackage("mathptmx");
 
usepackage("mathptmx");
 
defaultpen(linewidth(0.5));
 
defaultpen(linewidth(0.5));
 
size(5cm);
 
size(5cm);
 
defaultpen(fontsize(14pt));
 
defaultpen(fontsize(14pt));
−
label("<math>\textbf{Math}</math>", (2.1,3.7)--(3.9,3.7));
+
label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7));
−
label("<math>\textbf{Team}</math>", (2.1,3)--(3.9,3));
+
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);
 
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);
 +
 +
draw((3,4)--(4,3), red);
 +
draw((4,3)--(3,2), red);
 +
draw((3,2)--(2,3), red);
 +
draw((2,3)--(3,4), red);
 +
</asy>
 +
 +
We see these lines split the figure into five squares with side length <math>\sqrt2</math>. Thus, the area is <math>5\cdot\left(\sqrt2\right)^2=5\cdot 2 = \boxed{\textbf{(A) } 10}</math>.
 +
 +
~pog ~wamofan
 +
 +
==Solution 2==
 +
 +
There are <math>5</math> lattice points in the interior of the logo and <math>12</math> lattice points on the boundary of the logo. Because of Pick's Theorem, the area of the logo is <math>5+\frac{12}{2}-1=\boxed{\textbf{(A) } 10}</math>.
 +
 +
~MathFun1000
 +
 +
==Solution 3==
 +
 +
Notice that the area of the figure is equal to the area of the <math>4 \times 4</math> square subtracted by the <math>12</math> triangles that are half the area of each square, which is <math>1</math>. The total area of the triangles not in the figure is <math>12 \cdot \frac{1}{2} = 6</math>, so the answer is <math>16-6 = \boxed{\textbf{(A) } 10}</math>.
 +
 +
~hh99754539
 +
 +
==Solution 4==
 +
Draw the following four lines as shown:
 +
 +
<asy>
 +
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((0,0)--(6,0), gray);
Line 29: Line 107:
 
draw((6,0)--(6,6), gray);
 
draw((6,0)--(6,6), gray);
  
−
[/asy]
+
draw((2,4)--(4,4), red);
 +
draw((4,4)--(4,2), red);
 +
draw((4,2)--(2,2), red);
 +
draw((2,2)--(2,4), red);
 +
</asy>
  
−
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15</math>
+
The area of the big square is <math>4</math>, and the area of each triangle is <math>0.5</math>. There are <math>12</math> of these triangles, so the total area of all the triangles is <math>0.5\cdot12=6</math>. Therefore, the area of the entire figure is <math>4+6=\boxed{\textbf{(A) } 10}</math>.
 +
 
 +
~RocketScientist
 +
 
 +
==Solution 5 (Shoelace Theorem)==
 +
The coordinates are <math>(1,2), (2,1), (3,2), (4,1), (5,2), (4,3), (5,4), (4,5), (3,4), (2,5), (1,4), (2,3)</math>
 +
Use the [[Shoelace Theorem]] to get <math>\boxed{\textbf{(A)} ~10}</math>.
 +
 
 +
==Solution 6 (Quick) ==
 +
If the triangles are rearranged such that the gaps are filled, there would be a <math>4</math> by <math>2</math> rectangle, and two <math>1</math> by <math>1</math> squares are present. Thus, the answer is <math>\boxed{\textbf{(A)} ~10}</math>.
 +
 
 +
~peelybonehead
 +
 
 +
==Video Solution 1 by Math-X (First understand the problem!!!)==
 +
https://youtu.be/oUEa7AjMF2A?si=7nqtNywjcJi2uIf7&t=62
 +
 
 +
~Math-X
 +
 
 +
==Video Solution 2 (HOW TO CREATIVELY THINK!!!)==
 +
https://youtu.be/8TyfWyTOxLI
 +
 
 +
~Education, the Study of Everything
 +
 
 +
==Video Solution 3==
 +
https://www.youtube.com/watch?v=Ij9pAy6tQSg
 +
~Interstigation
 +
 
 +
==Video Solution 4==
 +
https://youtu.be/mKAqJxZMKTM
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution 5 ==
 +
https://youtu.be/Q0R6dnIO95Y
 +
 
 +
~STEMbreezy
 +
 
 +
==Video Solution 6 ==
 +
https://www.youtube.com/watch?v=pGpDR0hm6qs
 +
 
 +
~harungurcan
 +
 
 +
==See Also==
 +
{{AMC8 box|year=2022|before=First Problem|num-a=2}}
 +
{{MAA Notice}}

Latest revision as of 12:22, 21 December 2023

Problem

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?

[asy] 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); [/asy]

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

Solution 1

Draw the following four lines as shown: [asy] 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); draw((3,4)--(4,3), red); draw((4,3)--(3,2), red); draw((3,2)--(2,3), red); draw((2,3)--(3,4), red); [/asy]

We see these lines split the figure into five squares with side length $\sqrt2$. Thus, the area is $5\cdot\left(\sqrt2\right)^2=5\cdot 2 = \boxed{\textbf{(A) } 10}$.

~pog ~wamofan

Solution 2

There are $5$ lattice points in the interior of the logo and $12$ lattice points on the boundary of the logo. Because of Pick's Theorem, the area of the logo is $5+\frac{12}{2}-1=\boxed{\textbf{(A) } 10}$.

~MathFun1000

Solution 3

Notice that the area of the figure is equal to the area of the $4 \times 4$ square subtracted by the $12$ triangles that are half the area of each square, which is $1$. The total area of the triangles not in the figure is $12 \cdot \frac{1}{2} = 6$, so the answer is $16-6 = \boxed{\textbf{(A) } 10}$.

~hh99754539

Solution 4

Draw the following four lines as shown:

[asy] 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); draw((2,4)--(4,4), red); draw((4,4)--(4,2), red); draw((4,2)--(2,2), red); draw((2,2)--(2,4), red); [/asy]

The area of the big square is $4$, and the area of each triangle is $0.5$. There are $12$ of these triangles, so the total area of all the triangles is $0.5\cdot12=6$. Therefore, the area of the entire figure is $4+6=\boxed{\textbf{(A) } 10}$.

~RocketScientist

Solution 5 (Shoelace Theorem)

The coordinates are $(1,2), (2,1), (3,2), (4,1), (5,2), (4,3), (5,4), (4,5), (3,4), (2,5), (1,4), (2,3)$ Use the Shoelace Theorem to get $\boxed{\textbf{(A)} ~10}$.

Solution 6 (Quick)

If the triangles are rearranged such that the gaps are filled, there would be a $4$ by $2$ rectangle, and two $1$ by $1$ squares are present. Thus, the answer is $\boxed{\textbf{(A)} ~10}$.

~peelybonehead

Video Solution 1 by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=7nqtNywjcJi2uIf7&t=62

~Math-X

Video Solution 2 (HOW TO CREATIVELY THINK!!!)

https://youtu.be/8TyfWyTOxLI

~Education, the Study of Everything

Video Solution 3

https://www.youtube.com/watch?v=Ij9pAy6tQSg ~Interstigation

Video Solution 4

https://youtu.be/mKAqJxZMKTM

~savannahsolver

Video Solution 5

https://youtu.be/Q0R6dnIO95Y

~STEMbreezy

Video Solution 6

https://www.youtube.com/watch?v=pGpDR0hm6qs

~harungurcan

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_8_Problems/Problem_1&oldid=208031"