Difference between revisions of "2022 AMC 10A Problems/Problem 18"

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.

Problem

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself?

$\textbf{(A) } 359 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

Solution 1

Let $P=(r,\theta)$ be a point in polar coordinates, where $\theta$ is in degrees.

Rotating $P$ by $k^{\circ}$ counterclockwise around the origin gives the transformation $(r,\theta)\rightarrow(r,\theta+k^{\circ}).$ Reflecting $P$ across the $y$ -axis gives the transformation $(r,\theta)\rightarrow(r,180^{\circ}-\theta).$ Note that $\begin{align*} T_k(P)&=(r,180^{\circ}-\theta-k^{\circ}), \\ T_{k+1}(T_k(P)) &= (r,\theta -1^{\circ}). \end{align*}$ We start with $(1,0^{\circ})$ in polar coordinates. For the sequence of transformations $T_1, T_2, T_3, \cdots, T_k,$ it follows that

After $T_1,$ we have $(1,179^{\circ}).$

After $T_2,$ we have $(1,-1^{\circ}).$

After $T_3,$ we have $(1,178^{\circ}).$

After $T_4,$ we have $(1,-2^{\circ}).$

After $T_5,$ we have $(1,177^{\circ}).$

After $T_6,$ we have $(1,-3^{\circ}).$

...

After $T_{2k-1},$ we have $(1,180^{\circ}-k^{\circ}).$

After $T_{2k},$ we have $(1,-k^{\circ}).$

The least such positive integer $k$ is $180.$ Therefore, the least such positive integer $n$ is $2k-1=\boxed{\textbf{(A) } 359}.$

~MRENTHUSIASM

Solution 2

Note that since we're reflecting across the $y$ -axis, if the point ever makes it to $(-1,0)$ then it will flip back to the original point. Note that after $T_1$ the point will be $1$ degree clockwise from the negative $x$ -axis. Applying $T_2$ will rotate it to be $1$ degree counterclockwise from the negative $x$ -axis, and then flip it so that it is $1$ degree clockwise from the positive $x$ -axis. Therefore, after every $2$ transformations, the point rotates $1$ degree clockwise. To rotate it so that it will rotate $179$ degrees clockwise will require $179 \cdot 2 = 358$ transformations. Then finally on the last transformation, it will rotate on to $(-1,0)$ and then flip back to its original position. Therefore, the answer is $358+1 = 359 = \boxed{\textbf{(A) } 359}$ .

~KingRavi

Solution 3

In degrees:

Starting with $n=0$ , the sequence goes ${0}\rightarrow {179}\rightarrow {359}\rightarrow {178}\rightarrow {358}\rightarrow {177}\rightarrow {357}\rightarrow\cdots.$

We see that it takes $2$ steps to downgrade the point by $1^{\circ}$ . Since the $1$ st point in the sequence is ${179}$ , the answer is $1+2(179)=\boxed{\textbf{(A) } 359}.$

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, $359$ first, and we get that it turns $\frac{(1+359)359}{2} = 180n$ , where $n$ is an integer. Thus, the point would be rotated to $(-1,0)$ . We may be tempted to dismiss this option but we haven't considered the reflections. Each reflection acts as a $180^{\circ}$ rotation, so every two reflections cancel. However, $359$ is odd so we have to reflect $(-1,0)$ , taking us to $(1,0)$ , which is what we want. Thus we get $\boxed{\textbf{(A) } 359}$ .

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 $z = \exp (i\theta)$ , starting with $z=1+0i=\exp(i0)$ . Then, we can describe rotations and reflections about the $y$ -axis using these formulas: $\text{Rotation}(\exp(i\theta), k): \exp(i\theta) \mapsto \exp(i(\theta + k))$ $\text{Reflection}(\exp(i\theta)): \exp(i\theta) \mapsto \exp i (180^\circ - \theta) = -\exp(i(-\theta))$

If we apply two successive iterations, we see a simplification:

$T_k(\exp(i\theta)) = -\exp(-i(\theta+k))$

\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 $T_1(\exp i0) = \exp 179^\circ$ . Thus, the point $(1,0)$ gets sent back to when all double iterations after $1$ cancel $179^\circ$ . $179-\left(\frac{n-1}{2}\right)=0$ so $n = 1 + 2 \cdot 179 = \boxed{\textbf{(A) }359}$ .

Video Solution

https://youtu.be/QQrsKTErJn8

~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

@@ Line 1: / Line 1: @@
-{{duplicate|[[2022 AMC 10A Problems/Problem 1|2022 AMC 10A #1]] and [[2022 AMC 12A Problems/Problem 1|2022 AMC 12A #1]]}}
+{{duplicate|[[2022 AMC 10A Problems/Problem 18|2022 AMC 10A #18]] and [[2022 AMC 12A Problems/Problem 18|2022 AMC 12A #18]]}}
 ==Problem==
@@ Line 9: / Line 9: @@
 ==Solution 1==
-Let <math>P=(r,\theta)</math> be a point in polar coordinates. Rotating <math>P</math> by <math>k^{\circ}</math> counterclockwise around the origin gives the transformation <math>(r,\theta)\rightarrow(r,\theta+k^{\circ}).</math> Reflecting <math>P</math> across the <math>y</math>-axis gives the transformation <math>(r,\theta)\rightarrow(r,180^{\circ}-\theta).</math>
+Let <math>P=(r,\theta)</math> be a point in polar coordinates, where <math>\theta</math> is in degrees.
+Rotating <math>P</math> by <math>k^{\circ}</math> counterclockwise around the origin gives the transformation <math>(r,\theta)\rightarrow(r,\theta+k^{\circ}).</math> Reflecting <math>P</math> across the <math>y</math>-axis gives the transformation <math>(r,\theta)\rightarrow(r,180^{\circ}-\theta).</math> Note that
+<cmath>\begin{align*}
+T_k(P)&=(r,180^{\circ}-\theta-k^{\circ}), \\
+T_{k+1}(T_k(P)) &= (r,\theta -1^{\circ}).
+\end{align*}</cmath>
+We start with <math>(1,0^{\circ})</math> in polar coordinates. For the sequence of transformations <math>T_1, T_2, T_3, \cdots, T_k,</math> it follows that
+* After <math>T_1,</math> we have <math>(1,179^{\circ}).</math>
+* After <math>T_2,</math> we have <math>(1,-1^{\circ}).</math>
+* After <math>T_3,</math> we have <math>(1,178^{\circ}).</math>
+* After <math>T_4,</math> we have <math>(1,-2^{\circ}).</math>
+* After <math>T_5,</math> we have <math>(1,177^{\circ}).</math>
+* After <math>T_6,</math> we have <math>(1,-3^{\circ}).</math>
+* ...
+* After <math>T_{2k-1},</math> we have <math>(1,180^{\circ}-k^{\circ}).</math>
+* After <math>T_{2k},</math> we have <math>(1,-k^{\circ}).</math>
+The least such positive integer <math>k</math> is <math>180.</math> Therefore, the least such positive integer <math>n</math> is <math>2k-1=\boxed{\textbf{(A) } 359}.</math>
 ~MRENTHUSIASM
 ==Solution 2==
-Note that since we're reflecting across the <math>y</math>-axis, if the point ever makes it to <math>(-1,0)</math> then it will flip back to the original point. Note that after <math>T_1</math> the point will be <math>1</math> degree clockwise from the negative <math>x</math>-axis. Applying <math>T_2</math> will rotate it to be <math>1</math> degree counterclockwise from the negative <math>x</math>-axis, and then flip it so that it is <math>1</math> degree clockwise from the positive <math>x</math>-axis. Therefore, after every <math>2</math> transformations, the point rotates <math>1</math> degree clockwise. To rotate it so that it will rotate <math>179</math> degrees clockwise will require <math>179 \cdot 2 = 358</math> transformations. Then finally on the last transformation, it will rotate on to <math>(-1,0)</math> and then flip back to it's original position. Therefore, the answer is <math>358+1 = 359 = \boxed{A}</math>
+Note that since we're reflecting across the <math>y</math>-axis, if the point ever makes it to <math>(-1,0)</math> then it will flip back to the original point. Note that after <math>T_1</math> the point will be <math>1</math> degree clockwise from the negative <math>x</math>-axis. Applying <math>T_2</math> will rotate it to be <math>1</math> degree counterclockwise from the negative <math>x</math>-axis, and then flip it so that it is <math>1</math> degree clockwise from the positive <math>x</math>-axis. Therefore, after every <math>2</math> transformations, the point rotates <math>1</math> degree clockwise. To rotate it so that it will rotate <math>179</math> degrees clockwise will require <math>179 \cdot 2 = 358</math> transformations. Then finally on the last transformation, it will rotate on to <math>(-1,0)</math> and then flip back to its original position. Therefore, the answer is <math>358+1 = 359 = \boxed{\textbf{(A) } 359}</math>.
 ~KingRavi
 ==Solution 3==
-Let <math>A_{n}</math> be the point <math>(\cos n^{\circ}, \sin n^{\circ})</math>.
+In degrees:
+Starting with <math>n=0</math>, the sequence goes <math>{0}\rightarrow {179}\rightarrow {359}\rightarrow {178}\rightarrow {358}\rightarrow {177}\rightarrow {357}\rightarrow\cdots.</math>
+We see that it takes <math>2</math> steps to downgrade the point by <math>1^{\circ}</math>. Since the <math>1</math>st point in the sequence is <math>{179}</math>, the answer is <math>1+2(179)=\boxed{\textbf{(A) } 359}.</math>
+==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, <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>.
+==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:
-Starting with <math>n=0</math>, the sequence goes <cmath>A_{0}\rightarrow A_{179}\rightarrow A_{359}\rightarrow A_{178}\rightarrow A_{358}\rightarrow A_{177}\rightarrow A_{357}\rightarrow\cdots</cmath>
+<cmath>
+T_k(\exp(i\theta)) = -\exp(-i(\theta+k))
+</cmath>
-We see that it takes <math>2</math> turns to downgrade the point by <math>1^{\circ}</math>. Since the fifth point in the sequence is <math>A_{177}</math>, the answer is <math>5+2(177)=\boxed{\textbf{(A)}~359}</math>
+\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==
@@ Line 30: / Line 87: @@
 ~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

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