Difference between revisions of "2020 AMC 10B Problems/Problem 23"
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== Problem == | == Problem == | ||
− | Square <math> | + | Square <math>ABCD</math> in the coordinate plane has vertices at the points <math>A(1,1), B(-1,1), C(-1,-1),</math> and <math>D(1,-1).</math> Consider the following four transformations: |
+ | <math>\quad\bullet\qquad</math> <math>L,</math> a rotation of <math>90^{\circ}</math> counterclockwise around the origin; | ||
− | + | <math>\quad\bullet\qquad</math> <math>R,</math> a rotation of <math>90^{\circ}</math> clockwise around the origin; | |
− | + | <math>\quad\bullet\qquad</math> <math>H,</math> a reflection across the <math>x</math>-axis; and | |
− | + | <math>\quad\bullet\qquad</math> <math>V,</math> a reflection across the <math>y</math>-axis. | |
− | |||
− | |||
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying <math>R</math> and then <math>V</math> would send the vertex <math>A</math> at <math>(1,1)</math> to <math>(-1,-1)</math> and would send the vertex <math>B</math> at <math>(-1,1)</math> to itself. How many sequences of <math>20</math> transformations chosen from <math>\{L, R, H, V\}</math> will send all of the labeled vertices back to their original positions? (For example, <math>R, R, V, H</math> is one sequence of <math>4</math> transformations that will send the vertices back to their original positions.) | Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying <math>R</math> and then <math>V</math> would send the vertex <math>A</math> at <math>(1,1)</math> to <math>(-1,-1)</math> and would send the vertex <math>B</math> at <math>(-1,1)</math> to itself. How many sequences of <math>20</math> transformations chosen from <math>\{L, R, H, V\}</math> will send all of the labeled vertices back to their original positions? (For example, <math>R, R, V, H</math> is one sequence of <math>4</math> transformations that will send the vertices back to their original positions.) | ||
− | <math>\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ | + | <math>\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}</math> |
== Solution 1 == | == Solution 1 == | ||
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Next, even number of reflections mean that there must be even number of rotations since their sum is even. Even rotations result only in the original position or <math>180^{\circ}</math> rotation of it. | Next, even number of reflections mean that there must be even number of rotations since their sum is even. Even rotations result only in the original position or <math>180^{\circ}</math> rotation of it. | ||
− | Since rotation <math>R</math> and rotation <math>L</math> cancels each other out, the difference between the numbers of them define the final position. The probability of the transformation returning the vertices to the | + | Since rotation <math>R</math> and rotation <math>L</math> cancels each other out, the difference between the numbers of them define the final position. The probability of the transformation returning the vertices to the original position given that there are even number of rotations is equivalent to the probability that |
<center><math>|n(R)-n(L)|\equiv0\pmod{4}</math> when <math>|n(H)-n(V)|\equiv0\pmod{4}</math></center> | <center><math>|n(R)-n(L)|\equiv0\pmod{4}</math> when <math>|n(H)-n(V)|\equiv0\pmod{4}</math></center> | ||
Line 57: | Line 56: | ||
Notice that any pair of two of these transformations either swaps the x and y-coordinates, negates the x and y-coordinates, swaps and negates the x and y-coordinates, or leaves the original unchanged. Furthermore, notice that for each of these results, if we apply another pair of transformations, one of these results will happen again, and with equal probability. Therefore, no matter what state after we apply the first <math>19</math> pairs of transformations, there is a <math>\frac14</math> chance the last pair of transformations will return the figure to its original position. Therefore, the answer is <math>\frac{4^{20}}4 = 4^{19} = \boxed{\textbf{(C) }2^{38}}</math> | Notice that any pair of two of these transformations either swaps the x and y-coordinates, negates the x and y-coordinates, swaps and negates the x and y-coordinates, or leaves the original unchanged. Furthermore, notice that for each of these results, if we apply another pair of transformations, one of these results will happen again, and with equal probability. Therefore, no matter what state after we apply the first <math>19</math> pairs of transformations, there is a <math>\frac14</math> chance the last pair of transformations will return the figure to its original position. Therefore, the answer is <math>\frac{4^{20}}4 = 4^{19} = \boxed{\textbf{(C) }2^{38}}</math> | ||
− | == Video Solution 1 == | + | == Solution 5 == |
+ | For each transformation: | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li>Each labeled vertex will move to an adjacent position.</li><p> | ||
+ | <li>The labeled vertices will maintain the consecutive order <math>ABCD</math> in either direction (clockwise or counterclockwise).</li><p> | ||
+ | <li><math>L</math> and <math>R</math> will retain the direction of the labeled vertices, but <math>H</math> and <math>V</math> will alter the direction of the labeled vertices.</li><p> | ||
+ | </ol> | ||
+ | After the <math>19</math>th transformation, vertex <math>A</math> will be at either <math>(1,-1)</math> or <math>(-1,1).</math> All possible configurations of the labeled vertices are shown below: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | unitsize(7mm); | ||
+ | label("$A$",(1,0)); | ||
+ | label("$B$",(1,1)); | ||
+ | label("$C$",(0,1)); | ||
+ | label("$D$",(0,0)); | ||
+ | |||
+ | label("$C$",(5,0)); | ||
+ | label("$D$",(5,1)); | ||
+ | label("$A$",(4,1)); | ||
+ | label("$B$",(4,0)); | ||
+ | |||
+ | label("$A$",(9,0)); | ||
+ | label("$D$",(9,1)); | ||
+ | label("$C$",(8,1)); | ||
+ | label("$B$",(8,0)); | ||
+ | |||
+ | label("$C$",(13,0)); | ||
+ | label("$B$",(13,1)); | ||
+ | label("$A$",(12,1)); | ||
+ | label("$D$",(12,0)); | ||
+ | |||
+ | label("\textbf{Configuration}",(-5,0.5)); | ||
+ | label("\textbf{The 20th}",(-5,-1.5)); | ||
+ | label("\textbf{Transformation}",(-5,-2.25)); | ||
+ | label("$L$",(0.5,-1.875)); | ||
+ | label("$R$",(4.5,-1.875)); | ||
+ | label("$H$",(8.5,-1.875)); | ||
+ | label("$V$",(12.5,-1.875)); | ||
+ | </asy> | ||
+ | Each sequence of <math>19</math> transformations generates one valid sequence of <math>20</math> transformations. Therefore, the answer is <math>4^{19}=\boxed{\textbf{(C)}\ 2^{38}}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Video Solutions== | ||
+ | === Video Solution 1 === | ||
https://www.youtube.com/watch?v=XZs9DHg6cx0 ~ MathEx | https://www.youtube.com/watch?v=XZs9DHg6cx0 ~ MathEx | ||
− | + | ===Video Solution 2 by the Beauty Of Math=== | |
− | ==Video Solution 2 by the Beauty Of Math== | + | https://youtu.be/Bl2kn9oVxQ8?t=348 |
− | https://youtu.be/Bl2kn9oVxQ8 | ||
~icematrix | ~icematrix |
Latest revision as of 20:41, 5 June 2021
- The following problem is from both the 2020 AMC 10B #23 and 2020 AMC 12B #19, so both problems redirect to this page.
Contents
Problem
Square in the coordinate plane has vertices at the points and Consider the following four transformations:
a rotation of counterclockwise around the origin;
a rotation of clockwise around the origin;
a reflection across the -axis; and
a reflection across the -axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying and then would send the vertex at to and would send the vertex at to itself. How many sequences of transformations chosen from will send all of the labeled vertices back to their original positions? (For example, is one sequence of transformations that will send the vertices back to their original positions.)
Solution 1
Let (+) denote counterclockwise/starting orientation and (-) denote clockwise orientation. Let 1,2,3, and 4 denote which quadrant A is in.
Realize that from any odd quadrant and any orientation, the 4 transformations result in some permutation of .
The same goes that from any even quadrant and any orientation, the 4 transformations result in some permutation of .
We start our first 19 moves by doing whatever we want, 4 choices each time. Since 19 is odd, we must end up on an even quadrant.
As said above, we know that exactly one of the four transformations will give us , and we must use that transformation.
Thus
Solution 2
Hopefully, someone will think of a better one, but here is an indirect answer, use only if you are really desperate. moves can be made, and each move have choices, so a total of moves. First, after the moves, Point A can only be in first quadrant or third quadrant . Only the one in the first quadrant works, so divide by . Now, C must be in the opposite quadrant as A. B can be either in the second () or fourth quadrant () , but we want it to be in the second quadrant, so divide by again. Now as A and B satisfy the conditions, C and D will also be at their original spot. . The answer is ~Kinglogic
Solution 3
The total number of sequence is .
Note that there can only be even number of reflections since they result in the same anti-clockwise orientation of the verices . Therefore, the probability of having the same anti-clockwise orientation with the original arrangement after the transformation is .
Next, even number of reflections mean that there must be even number of rotations since their sum is even. Even rotations result only in the original position or rotation of it.
Since rotation and rotation cancels each other out, the difference between the numbers of them define the final position. The probability of the transformation returning the vertices to the original position given that there are even number of rotations is equivalent to the probability that
which is again, .
Therefore, ~joshuamh111
Solution 4
Notice that any pair of two of these transformations either swaps the x and y-coordinates, negates the x and y-coordinates, swaps and negates the x and y-coordinates, or leaves the original unchanged. Furthermore, notice that for each of these results, if we apply another pair of transformations, one of these results will happen again, and with equal probability. Therefore, no matter what state after we apply the first pairs of transformations, there is a chance the last pair of transformations will return the figure to its original position. Therefore, the answer is
Solution 5
For each transformation:
- Each labeled vertex will move to an adjacent position.
- The labeled vertices will maintain the consecutive order in either direction (clockwise or counterclockwise).
- and will retain the direction of the labeled vertices, but and will alter the direction of the labeled vertices.
After the th transformation, vertex will be at either or All possible configurations of the labeled vertices are shown below: Each sequence of transformations generates one valid sequence of transformations. Therefore, the answer is
~MRENTHUSIASM
Video Solutions
Video Solution 1
https://www.youtube.com/watch?v=XZs9DHg6cx0 ~ MathEx
Video Solution 2 by the Beauty Of Math
https://youtu.be/Bl2kn9oVxQ8?t=348
~icematrix
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 |
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.