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Difference between revisions of "2020 AMC 10B Problems/Problem 13"

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==Problem==
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== Problem ==
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Andy the Ant lives on a coordinate plane and is currently at <math>(-20, 20)</math> facing east (that is, in the positive <math>x</math>-direction). Andy moves <math>1</math> unit and then turns <math>90^{\circ}</math> degrees left. From there, Andy moves <math>2</math> units (north) and then turns <math>90^{\circ}</math> degrees left. He then moves <math>3</math> units (west) and again turns <math>90^{\circ}</math> degrees left. Andy continues his progress, increasing his distance each time by <math>1</math> unit and always turning left. What is the location of the point at which Andy makes the <math>2020</math>th left turn?
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Andy the Ant lives on a coordinate plane and is currently at <math>(-20, 20)</math> facing east (that is, in the positive <math>x</math>-direction). Andy moves <math>1</math> unit and then turns <math>90^{\circ}</math> left. From there, Andy moves <math>2</math> units (north) and then turns <math>90^{\circ}</math> left. He then moves <math>3</math> units (west) and again turns <math>90^{\circ}</math> left. Andy continues his progress, increasing his distance each time by <math>1</math> unit and always turning left. What is the location of the point at which Andy makes the <math>2020</math>th left turn?
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<math>\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)</math>
 
<math>\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)</math>
  
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==Solution 1==
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== Solution 1 (Associative Property) ==
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You can find that every four moves both coordinates decrease by 2. Therefore, both coordinates need to decrease by two 505 times. You subtract, giving you <math>\boxed{\textbf{(B) } \text{(-1020,-990)}}</math> -happykeeper
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Andy makes a total of <math>2020</math> moves: <math>1010</math> horizontal (<math>505</math> left and <math>505</math> right) and <math>1010</math> vertical (<math>505</math> up and <math>505</math> down).
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The <math>x</math>-coordinate of Andy's final position is <cmath>-20+\overbrace{\underbrace{1-3}_{-2}+\underbrace{5-7}_{-2}+\underbrace{9-11}_{-2}+\cdots+\underbrace{2017-2019}_{-2}}^{\text{1010 terms, 505 pairs}}=-20-2\cdot505=-1030.</cmath>
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The <math>y</math>-coordinate of Andy's final position is <cmath>20+\overbrace{\underbrace{2-4}_{-2}+\underbrace{6-8}_{-2}+\underbrace{10-12}_{-2}+\cdots+\underbrace{2018-2020}_{-2}}^{\text{1010 terms, 505 pairs}}=20-2\cdot505=-990.</cmath>
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Together, we have <math>(x,y)=\boxed{\textbf{(B)}\ (-1030, -990)}.</math>
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~MRENTHUSIASM
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 +
== Solution 2 (Pattern) ==
 +
You can find that every four moves both coordinates decrease by <math>2.</math> Therefore, both coordinates need to decrease by two <math>505</math> times. You subtract, giving you the answer of <math>\boxed{\textbf{(B)}\ (-1030, -990)}.</math>
 +
 
 +
~happykeeper
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 +
== Solution 3 (Pattern) ==
 +
 
 +
Let's first mark the first few points Andy will arrive at:
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Starting Point (<math>0^{\text{th}}</math> move): <math>(-20,20)</math>
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<math>1^{\text{st}}</math> move: <math>(-19,20)</math>
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<math>2^{\text{nd}}</math> move: <math>(-19,22)</math>
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<math>3^{\text{rd}}</math> move: <math>(-22,22)</math>
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<math>4^{\text{th}}</math> move: <math>(-22,18)</math>
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<math>5^{\text{th}}</math> move: <math>(-17,18)</math>
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<math>6^{\text{th}}</math> move: <math>(-17,24)</math>
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<math>7^{\text{th}}</math> move: <math>(-24, 24)</math>
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In the <math>3^{\text{rd}}</math> move Andy lands on <math>(-22,22)</math>, in the <math>7^{\text{th}}</math> move, Andy lands on <math>(-24, 24)</math>.
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There is a pattern, for every <math>4</math> moves (starting from the <math>3^{\text{rd}}</math> move), Andy will arrive on a coordinate in the form of <math>(-2n, 2n)</math>. From this we can deduce:
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<math>(1) \ 3^{\text{rd}}</math> move: <math>(-22,22)</math>
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<math>(2) \ 7^{\text{th}}</math> move: <math>(-24, 24)</math>
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<math>(3) \ 11^{\text{th}}</math> move: <math>(-26, 26)</math>
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<math>(4) \ 15^{\text{th}}</math> move: <math>(-28, 28)</math>
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<math>(n) \ (4n-1)^{\text{th}}</math> move: <math>(-20-2n, 20+2n)</math>
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We have <math>2019 = 4 \cdot 505 -1</math>, for which <math>n = 505</math> and <math>20 + 2 \cdot 505 = 1030</math>. So, on the <math>2019^{\text{th}}</math> move Andy is at <math>(-1030, 1030)</math>.
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Because the problem asks for the <math>2020^{\text{th}}</math> move, <math>1030-2020=-990</math>, on the <math>2020^{\text{th}}</math> move, Andy will be on <math>\boxed{\textbf{(B)}\ (-1030, -990)}</math>.
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~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
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== Solution 4 (Answer Choices) ==
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Drawing the spiral out, you obtain that every 4th turn occurs on the bottom left corner, whose coordinates difference stay the same(40):
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 +
(-20, 20)
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(-22, 18)
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(-24, 16)
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...
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So the X and Y coordinates will always have a difference of 40.
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Now looking at the answer choices, only answer choice B's X and Y coordinates have a difference of 40, there (B) is the answer. :)
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~Alan
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==Video Solution (HOW TO CRITICALLY THINK!!!)==
 +
https://youtu.be/gbnWUskoOuU
 +
 
 +
~Education, the Study of Everything
  
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==See Also==
 
−
2015 AMC 10B Problem 24
 
−
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_24
 
  
−
==Video Solution==
 
−
https://youtu.be/t6yjfKXpwDs
 
  
−
~IceMatrix
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== Video Solution ==
 +
https://youtu.be/t6yjfKXpwDs?t=413
  
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==See Also==
+
== Similar Problem ==
 +
2015 AMC 10B Problem 24
 +
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_24
  
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== See Also ==
 
{{AMC10 box|year=2020|ab=B|num-b=12|num-a=14}}
 
{{AMC10 box|year=2020|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:42, 30 October 2024

Problem

Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn?

$\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)$

Solution 1 (Associative Property)

Andy makes a total of $2020$ moves: $1010$ horizontal ($505$ left and $505$ right) and $1010$ vertical ($505$ up and $505$ down).

The $x$-coordinate of Andy's final position is \[-20+\overbrace{\underbrace{1-3}_{-2}+\underbrace{5-7}_{-2}+\underbrace{9-11}_{-2}+\cdots+\underbrace{2017-2019}_{-2}}^{\text{1010 terms, 505 pairs}}=-20-2\cdot505=-1030.\] The $y$-coordinate of Andy's final position is \[20+\overbrace{\underbrace{2-4}_{-2}+\underbrace{6-8}_{-2}+\underbrace{10-12}_{-2}+\cdots+\underbrace{2018-2020}_{-2}}^{\text{1010 terms, 505 pairs}}=20-2\cdot505=-990.\] Together, we have $(x,y)=\boxed{\textbf{(B)}\ (-1030, -990)}.$

~MRENTHUSIASM

Solution 2 (Pattern)

You can find that every four moves both coordinates decrease by $2.$ Therefore, both coordinates need to decrease by two $505$ times. You subtract, giving you the answer of $\boxed{\textbf{(B)}\ (-1030, -990)}.$

~happykeeper

Solution 3 (Pattern)

Let's first mark the first few points Andy will arrive at:

Starting Point ($0^{\text{th}}$ move): $(-20,20)$

$1^{\text{st}}$ move: $(-19,20)$

$2^{\text{nd}}$ move: $(-19,22)$

$3^{\text{rd}}$ move: $(-22,22)$

$4^{\text{th}}$ move: $(-22,18)$

$5^{\text{th}}$ move: $(-17,18)$

$6^{\text{th}}$ move: $(-17,24)$

$7^{\text{th}}$ move: $(-24, 24)$

In the $3^{\text{rd}}$ move Andy lands on $(-22,22)$, in the $7^{\text{th}}$ move, Andy lands on $(-24, 24)$.

There is a pattern, for every $4$ moves (starting from the $3^{\text{rd}}$ move), Andy will arrive on a coordinate in the form of $(-2n, 2n)$. From this we can deduce:

$(1) \ 3^{\text{rd}}$ move: $(-22,22)$

$(2) \ 7^{\text{th}}$ move: $(-24, 24)$

$(3) \ 11^{\text{th}}$ move: $(-26, 26)$

$(4) \ 15^{\text{th}}$ move: $(-28, 28)$

$(n) \ (4n-1)^{\text{th}}$ move: $(-20-2n, 20+2n)$

We have $2019 = 4 \cdot 505 -1$, for which $n = 505$ and $20 + 2 \cdot 505 = 1030$. So, on the $2019^{\text{th}}$ move Andy is at $(-1030, 1030)$.

Because the problem asks for the $2020^{\text{th}}$ move, $1030-2020=-990$, on the $2020^{\text{th}}$ move, Andy will be on $\boxed{\textbf{(B)}\ (-1030, -990)}$.

~isabelchen

Solution 4 (Answer Choices)

Drawing the spiral out, you obtain that every 4th turn occurs on the bottom left corner, whose coordinates difference stay the same(40):

(-20, 20) (-22, 18) (-24, 16) ...

So the X and Y coordinates will always have a difference of 40.

Now looking at the answer choices, only answer choice B's X and Y coordinates have a difference of 40, there (B) is the answer. :)

~Alan

Video Solution (HOW TO CRITICALLY THINK!!!)

https://youtu.be/gbnWUskoOuU

~Education, the Study of Everything


Video Solution

https://youtu.be/t6yjfKXpwDs?t=413

Similar Problem

2015 AMC 10B Problem 24 https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_24

See Also

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 10 Problems and Solutions

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