2021 AIME I Problems/Problem 7

Revision as of 19:58, 11 March 2021 by Cellsecret (talk | contribs) (Solution)

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

Since $-1\leq\sin(x)\leq1$, $\sin(mx)+\sin(nx)=2$ means that each of $\sin(mx)$ and $\sin(nx)$ must be exactly $1$. Then $m$ and $n$ must be cycles away, or the difference between them must be multiple of $4$. If $m$ is $1$, then $n$ can be $5,9,13,17,21,25,29$. Like this, the table below can be listed:

Range of $m$ Number of Possible $n$'s
Case 1 $1 \leq m \leq 2$ $7$
Case 2 $3 \leq m \leq 6$ $6$
Case 3 $7 \leq m \leq 10$ $5$
Case 4 $11 \leq m \leq 14$ $4$
Case 5 $15 \leq m \leq 18$ $3$
Case 6 $19 \leq m \leq 22$ $2$
Case 7 $23 \leq m \leq 26$ $1$
Case 8 $27 \leq m \leq 30$ $0$

In total, there are $\boxed{62}$ possible solutions.

However the answer is $63$, where is the last possible solution?

~Interstigation

See also

2021 AIME I (ProblemsAnswer KeyResources)
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