Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution)
(→‎Solution)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
−
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:
 
−
 
−
{| class="wikitable" style="text-align:center;width:100%"
 
−
|-
 
−
|
 
−
! scope="col" | '''Range of <math>m</math>'''
 
−
! scope="col" | '''Number of Possible <math>n</math>'s'''
 
−
|-
 
−
! scope="row" | '''Case 1'''
 
−
| <math>1 \leq m \leq 2</math>
 
−
| <math>7</math>
 
−
|-
 
−
! scope="row" | '''Case 2'''
 
−
| <math>3 \leq m \leq 6</math>
 
−
| <math>6</math>
 
−
|-
 
−
! scope="row" | '''Case 3'''
 
−
| <math>7 \leq m \leq 10</math>
 
−
| <math>5</math>
 
−
|-
 
−
! scope="row" | '''Case 4'''
 
−
| <math>11 \leq m \leq 14</math>
 
−
| <math>4</math>
 
−
|-
 
−
! scope="row" | '''Case 5'''
 
−
| <math>15 \leq m \leq 18</math>
 
−
| <math>3</math>
 
−
|-
 
−
! scope="row" | '''Case 6'''
 
−
| <math>19 \leq m \leq 22</math>
 
−
| <math>2</math>
 
−
|-
 
−
! scope="row" | '''Case 7'''
 
−
| <math>23 \leq m \leq 26</math>
 
−
| <math>1</math>
 
−
|-
 
−
! scope="row" | '''Case 8'''
 
−
| <math>27 \leq m \leq 30</math>
 
−
| <math>0</math>
 
−
|-
 
−
|}
 
−
 
−
In total, there are <math>\boxed{062}</math> possible solutions.
 
−
 
−
However the answer is <math>063</math>, where is the last possible solution?
 
−
 
−
~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 19:02, 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AIME_I_Problems/Problem_7&oldid=149119"