2021 AIME I Problems/Problem 7

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

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
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2021 AIME I Problems/Problem 7

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