{"status": "success", "data": {"description_md": "A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \\alpha | + |\\gamma |$?\n\n$\\textbf{(A)} \\; 1 \\qquad \\textbf{(B)} \\; \\sqrt {2} \\qquad \\textbf{(C)} \\; 2 \\qquad \\textbf{(D)} \\; 2 \\sqrt {2} \\qquad \\textbf{(E)} \\; 4 \\qquad$<br>([[2008 AMC 12B Problems/Problem 19|Solution]])\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>A function  <span class=\"katex--inline\">f</span>  is defined by  <span class=\"katex--inline\">f(z) = (4 + i) z^2 + \\alpha z + \\gamma</span>  for all complex numbers  <span class=\"katex--inline\">z</span> , where  <span class=\"katex--inline\">\\alpha</span>  and  <span class=\"katex--inline\">\\gamma</span>  are complex numbers and  <span class=\"katex--inline\">i^2 = - 1</span> . Suppose that  <span class=\"katex--inline\">f(1)</span>  and  <span class=\"katex--inline\">f(i)</span>  are both real. What is the smallest possible value of  <span class=\"katex--inline\">| \\alpha | + |\\gamma |</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} \\; 1 \\qquad \\textbf{(B)} \\; \\sqrt {2} \\qquad \\textbf{(C)} \\; 2 \\qquad \\textbf{(D)} \\; 2 \\sqrt {2} \\qquad \\textbf{(E)} \\; 4 \\qquad</span> <br/>([[2008 AMC 12B Problems/Problem 19|Solution]])</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": "2008 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/08_amc12B_p20", "prev": "/problem/08_amc12B_p18"}}