2022 AMC 12A Problems/Problem 17

Problem

Supppose $a$ is a real number such that the equation $a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}$ has more than one solution in the interval $(0, \pi)$ . The set of all such $a$ that can be written in the form $(p,q) \cup (q,r),$ where $p, q,$ and $r$ are real numbers with $p < q< r$ . What is $p+q+r$ ?

$\textbf{(A) } -4 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 4$

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2022 AMC 12A Problems/Problem 17

Problem