Difference between revisions of "2020 AMC 10B Problems/Problem 13"
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− | == Solution 3 == | + | == Solution 3 (Pattern) == |
Let's first mark the first few points Andy will arrive at: | Let's first mark the first few points Andy will arrive at: | ||
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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>. | 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>. | ||
− | ~isabelchen | + | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] |
== Video Solution == | == Video Solution == |
Revision as of 09:35, 28 December 2021
Contents
Problem
Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive
-direction). Andy moves
unit and then turns
left. From there, Andy moves
units (north) and then turns
left. He then moves
units (west) and again turns
left. Andy continues his progress, increasing his distance each time by
unit and always turning left. What is the location of the point at which Andy makes the
th left turn?
Solution 1
Andy makes a total of moves:
horizontal (
left and
right) and
vertical (
up and
down).
The -coordinate of Andy's final position is
The
-coordinate of Andy's final position is
Together, we have
~MRENTHUSIASM
Solution 2
You can find that every four moves both coordinates decrease by Therefore, both coordinates need to decrease by two
times. You subtract, giving you the answer of
~happykeeper
Solution 3 (Pattern)
Let's first mark the first few points Andy will arrive at:
Starting Point ( move):
move:
move:
move:
move:
move:
move:
move:
In the move Andy lands on
, in the
move, Andy lands on
.
There is a pattern, for every moves (starting from the
move), Andy will arrive on a coordinate in the form of
. From this we can deduce:
move:
move:
move:
move:
move:
We have , for which
and
. So, on the
move Andy is at
.
Because the problem asks for the move,
, on the
move, Andy will be on
.
Video Solution
~IceMatrix
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.