{"status": "success", "data": {"description_md": "Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\\pi]$ for which\n\n$$\\sin(x+y)\\leq \\sin(x)+\\sin(y)$$for every $x$ between $0$ and $\\pi$, inclusive?\n\n$\\textbf{(A) } y=0 \\qquad \\textbf{(B) } 0\\leq y\\leq \\frac{\\pi}{4} \\qquad \\textbf{(C) } 0\\leq y\\leq \\frac{\\pi}{2} \\qquad \\textbf{(D) } 0\\leq y\\leq \\frac{3\\pi}{4} \\qquad \\textbf{(E) } 0\\leq y\\leq \\pi$\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>Which of the following describes the largest subset of values of  <span class=\"katex--inline\">y</span>  within the closed interval  <span class=\"katex--inline\">[0,\\pi]</span>  for which</p>&#10;<p> <span class=\"katex--display\">\\sin(x+y)\\leq \\sin(x)+\\sin(y)</span> for every  <span class=\"katex--inline\">x</span>  between  <span class=\"katex--inline\">0</span>  and  <span class=\"katex--inline\">\\pi</span> , inclusive?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } y=0 \\qquad \\textbf{(B) } 0\\leq y\\leq \\frac{\\pi}{4} \\qquad \\textbf{(C) } 0\\leq y\\leq \\frac{\\pi}{2} \\qquad \\textbf{(D) } 0\\leq y\\leq \\frac{3\\pi}{4} \\qquad \\textbf{(E) } 0\\leq y\\leq \\pi</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": 2, "problem_name": "2018 AMC 12A Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12A_p10", "prev": "/problem/18_amc12A_p08"}}