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| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 18]] |
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| − | Let <math>T_k</math> be the transformation of the coordinate plane that first rotates the plane <math>k</math> degrees counter-clockwise around the origin and then reflects the plane across the <math>y</math>-axis. What is the least positive
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| − | integer <math>n</math> such that performing the sequence of transformations <math>T_1, T_2, T_3, \cdots, T_n</math> returns the point <math>(1,0)</math> back to itself?
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| − | ==Solution==
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| − | Let <math>A_{n}</math> be the point <math>(\cos n^{\circ}, \sin n^{\circ})</math>.
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| − | 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>
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| − | 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>
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| − | ==Video Solution by Professor Chen Education Palace==
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| − | https://youtu.be/QQrsKTErJn8
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| − | ==See also==
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| − | {{AMC12 box|year=2022|ab=A|num-b=17|num-a=19}}
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| − | {{AMC10 box|year=2022|ab=A|num-b=17|num-a=19}}
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| − | {{MAA Notice}}
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