{"status": "success", "data": {"description_md": "For complex numbers $u=a+bi$ and $v=c+di$, define the binary operation $\\otimes$ by$$u\\otimes v=ac+bdi.$$Suppose $z$ is a complex number such that $z\\otimes z=z^{2}+40$. What is $|z|$?\n\n$\\textbf{(A) }2\\qquad\\textbf{(B) }5\\qquad\\textbf{(C) }\\sqrt{5}\\qquad\\textbf{(D) }\\sqrt{10}\\qquad\\textbf{(E) }5\\sqrt{2}$\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>For complex numbers  <span class=\"katex--inline\">u=a+bi</span>  and  <span class=\"katex--inline\">v=c+di</span> , define the binary operation  <span class=\"katex--inline\">\\otimes</span>  by <span class=\"katex--display\">u\\otimes v=ac+bdi.</span> Suppose  <span class=\"katex--inline\">z</span>  is a complex number such that  <span class=\"katex--inline\">z\\otimes z=z^{2}+40</span> . What is  <span class=\"katex--inline\">|z|</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }2\\qquad\\textbf{(B) }5\\qquad\\textbf{(C) }\\sqrt{5}\\qquad\\textbf{(D) }\\sqrt{10}\\qquad\\textbf{(E) }5\\sqrt{2}</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": "2023 AMC 12B Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12B_p13", "prev": "/problem/23_amc12B_p11"}}