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Difference between revisions of "2021 AIME I Problems/Problem 7"

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==Solution==
 
==Solution==
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Since <math>-1\leq\sin(x)\leq1</math>, <math>\sin(mx)+\sin(nx)=2</math> means that each of <math>\sin(mx)</math> and <math>\sin(nx)</math> must be exactly <math>1</math>. Then <math>m</math> and <math>n</math> must be cycles away, or the difference between them must be multiple of <math>4</math>. If <math>m</math> is <math>1</math>, then <math>n</math> can be <math>5,9,13,17,21,25,29</math>. Like this, the table below can be listed:
 
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{| class="wikitable" style="text-align:center;width:100%"
 
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|-
 
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|
 
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! scope="col" | '''Range of <math>m</math>'''
 
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! scope="col" | '''Number of Possible <math>n</math>'s'''
 
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|-
 
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! scope="row" | '''Case 1'''
 
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| <math>1 \leq m \leq 2</math>
 
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| <math>7</math>
 
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|-
 
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! scope="row" | '''Case 2'''
 
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| <math>3 \leq m \leq 6</math>
 
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| <math>6</math>
 
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|-
 
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! scope="row" | '''Case 3'''
 
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| <math>7 \leq m \leq 10</math>
 
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| <math>5</math>
 
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|-
 
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! scope="row" | '''Case 4'''
 
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| <math>11 \leq m \leq 14</math>
 
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| <math>4</math>
 
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|-
 
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! scope="row" | '''Case 5'''
 
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| <math>15 \leq m \leq 18</math>
 
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| <math>3</math>
 
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|-
 
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! scope="row" | '''Case 6'''
 
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| <math>19 \leq m \leq 22</math>
 
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| <math>2</math>
 
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|-
 
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! scope="row" | '''Case 7'''
 
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| <math>23 \leq m \leq 26</math>
 
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| <math>1</math>
 
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|-
 
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! scope="row" | '''Case 8'''
 
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| <math>27 \leq m \leq 30</math>
 
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| <math>0</math>
 
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|}
 
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In total, there are <math>\boxed{062}</math> possible solutions.
 
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However the answer is <math>063</math>, where is the last possible solution?
 
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~Interstigation
 
  
 
==See also==
 
==See also==
 
{{AIME box|year=2021|n=I|num-b=6|num-a=8}}
 
{{AIME box|year=2021|n=I|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:02, 11 March 2021

Problem

Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying\[\sin(mx)+\sin(nx)=2.\]

Solution

See also

2021 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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