{"status": "success", "data": {"description_md": "Suppose $a$ is a real number such that the equation $$a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)}$$<br>has more than one solution in the interval $(0, \\pi)$. The set of all such $a$ that can be written<br>in the form $$(p,q) \\cup (q,r),$$<br>where $p, q,$ and $r$ are real numbers with $p < q< r$. What is $p+q+r$?\n\n$\\textbf{(A) } {-}4 \\qquad \\textbf{(B) } {-}1 \\qquad \\textbf{(C) } 0 \\qquad \\textbf{(D) } 1 \\qquad \\textbf{(E) } 4$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Suppose  <span class=\"katex--inline\">a</span>  is a real number such that the equation  <span class=\"katex--display\">a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)}</span> <br/>has more than one solution in the interval  <span class=\"katex--inline\">(0, \\pi)</span> . The set of all such  <span class=\"katex--inline\">a</span>  that can be written<br/>in the form  <span class=\"katex--display\">(p,q) \\cup (q,r),</span> <br/>where  <span class=\"katex--inline\">p, q,</span>  and  <span class=\"katex--inline\">r</span>  are real numbers with  <span class=\"katex--inline\">p &lt; q&lt; r</span> . What is  <span class=\"katex--inline\">p+q+r</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } {-}4 \\qquad \\textbf{(B) } {-}1 \\qquad \\textbf{(C) } 0 \\qquad \\textbf{(D) } 1 \\qquad \\textbf{(E) } 4</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2022 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12A_p18", "prev": "/problem/22_amc12A_p16"}}