Difference between revisions of "2022 AMC 10A Problems/Problem 18"
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==Solution 4 (Simple)== | ==Solution 4 (Simple)== | ||
− | We | + | We can consider the rotations and reflections separately. For the rotations, each rotation turns it by the next natural number. Thus the total number of degrees turned would be a triangle number. We test the smallest number, <math>359</math> first, and we get that it turns <math>\frac{(1+359)359}{2} = 180n</math>, where <math>n</math> is an integer. Thus, the point would be rotated to <math>(-1,0)</math>. We may be tempted to dismiss this option but we haven't considered the reflections. Each reflection acts as a <math>180^{\circ}</math> rotation, so every two reflections cancel. However, <math>359</math> is odd so we have to reflect <math>(-1,0)</math>, taking us to <math>(1,0)</math>, which is what we want. Thus we get <math>\boxed{\textbf{(A) } 359}</math>. |
− | This | + | ==Solution 5 (Complex)== |
+ | Rotations and reflections in 2D are very nice to describe using complex numbers in polar form. Reflections and rotations also don't affect the length (modulus or absolute value) of a complex number. This motivates us to set <math>z = \exp (i\theta)</math>, starting with <math>z=1+0i=\exp(i0)</math>. | ||
+ | Then, we can describe rotations and reflections about the <math>y</math>-axis using these formulas: | ||
+ | <cmath> | ||
+ | \text{Rotation}(\exp(i\theta), k): \exp(i\theta) \mapsto \exp(i(\theta + k)) | ||
+ | </cmath> | ||
+ | <cmath> | ||
+ | \text{Reflection}(\exp(i\theta)): \exp(i\theta) \mapsto \exp i (180^\circ - \theta) = -\exp(i(-\theta)) | ||
+ | </cmath> | ||
+ | |||
+ | If we apply two successive iterations, we see a simplification: | ||
+ | |||
+ | <cmath> | ||
+ | T_k(\exp(i\theta)) = -\exp(-i(\theta+k)) | ||
+ | </cmath> | ||
+ | |||
+ | \begin{aligned} | ||
+ | T_{k+1}(T_k(\exp(i\theta))) &= T_{k+1}(-\exp(i(-\theta-k)) | ||
+ | \ | ||
+ | &= --\exp(-i(-\theta-k+(k+1)) | ||
+ | \ | ||
+ | &= \exp i(\theta-1) | ||
+ | \end{aligned} | ||
+ | |||
+ | |||
+ | We also can calculate <math>T_1(\exp i0) = \exp 179^\circ</math>. Thus, the point <math>(1,0)</math> gets sent back to when all double iterations after <math>1</math> cancel <math>179^\circ</math>. <math>179-\left(\frac{n-1}{2}\right)=0</math> so <math>n = 1 + 2 \cdot 179 = \boxed{\textbf{(A) }359}</math>. | ||
==Video Solution== | ==Video Solution== |
Latest revision as of 10:19, 9 November 2024
- The following problem is from both the 2022 AMC 10A #18 and 2022 AMC 12A #18, so both problems redirect to this page.
Contents
[hide]Problem
Let be the transformation of the coordinate plane that first rotates the plane degrees counterclockwise around the origin and then reflects the plane across the -axis. What is the least positive integer such that performing the sequence of transformations returns the point back to itself?
Solution 1
Let be a point in polar coordinates, where is in degrees.
Rotating by counterclockwise around the origin gives the transformation Reflecting across the -axis gives the transformation Note that We start with in polar coordinates. For the sequence of transformations it follows that
- After we have
- After we have
- After we have
- After we have
- After we have
- After we have
- ...
- After we have
- After we have
The least such positive integer is Therefore, the least such positive integer is
~MRENTHUSIASM
Solution 2
Note that since we're reflecting across the -axis, if the point ever makes it to then it will flip back to the original point. Note that after the point will be degree clockwise from the negative -axis. Applying will rotate it to be degree counterclockwise from the negative -axis, and then flip it so that it is degree clockwise from the positive -axis. Therefore, after every transformations, the point rotates degree clockwise. To rotate it so that it will rotate degrees clockwise will require transformations. Then finally on the last transformation, it will rotate on to and then flip back to its original position. Therefore, the answer is .
~KingRavi
Solution 3
In degrees:
Starting with , the sequence goes
We see that it takes steps to downgrade the point by . Since the st point in the sequence is , the answer is
Solution 4 (Simple)
We can consider the rotations and reflections separately. For the rotations, each rotation turns it by the next natural number. Thus the total number of degrees turned would be a triangle number. We test the smallest number, first, and we get that it turns , where is an integer. Thus, the point would be rotated to . We may be tempted to dismiss this option but we haven't considered the reflections. Each reflection acts as a rotation, so every two reflections cancel. However, is odd so we have to reflect , taking us to , which is what we want. Thus we get .
Solution 5 (Complex)
Rotations and reflections in 2D are very nice to describe using complex numbers in polar form. Reflections and rotations also don't affect the length (modulus or absolute value) of a complex number. This motivates us to set , starting with . Then, we can describe rotations and reflections about the -axis using these formulas:
If we apply two successive iterations, we see a simplification:
We also can calculate . Thus, the point gets sent back to when all double iterations after cancel . so .
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution (Simple and Fun!!!)
https://youtu.be/7yAh4MtJ8a8?si=2UC_9X7DjkL8UW5C&t=4968
~Math-X
See also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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.