{"status": "success", "data": {"description_md": "Let $\\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\\mathcal{R}$?  \n\n$\\textbf{(A) } 13 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 15 \\qquad \\textbf{(D) } 16 \\qquad \\textbf{(E) } 17$\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>Let  <span class=\"katex--inline\">\\mathcal{R}</span>  be the region in the complex plane consisting of all complex numbers  <span class=\"katex--inline\">z</span>  that can be written as the sum of complex numbers  <span class=\"katex--inline\">z_1</span>  and  <span class=\"katex--inline\">z_2</span> , where  <span class=\"katex--inline\">z_1</span>  lies on the segment with endpoints  <span class=\"katex--inline\">3</span>  and  <span class=\"katex--inline\">4i</span> , and  <span class=\"katex--inline\">z_2</span>  has magnitude at most  <span class=\"katex--inline\">1</span> . What integer is closest to the area of  <span class=\"katex--inline\">\\mathcal{R}</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 13 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 15 \\qquad \\textbf{(D) } 16 \\qquad \\textbf{(E) } 17</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 13", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12A_p14", "prev": "/problem/22_amc12A_p12"}}