{"status": "success", "data": {"description_md": "For integers $a$, $b$, $c$, and $d$, let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$. Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2)) = g(f(4)) = 0$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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 integers <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, <span class=\"katex--inline\">c</span>, and <span class=\"katex--inline\">d</span>, let <span class=\"katex--inline\">f(x) = x^2 + ax + b</span> and <span class=\"katex--inline\">g(x) = x^2 + cx + d</span>. Find the number of ordered triples <span class=\"katex--inline\">(a,b,c)</span> of integers with absolute values not exceeding <span class=\"katex--inline\">10</span> for which there is an integer <span class=\"katex--inline\">d</span> such that <span class=\"katex--inline\">g(f(2)) = g(f(4)) = 0</span>.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": 5, "problem_name": "2020 AIME I Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/20_aime_I_p12", "prev": "/problem/20_aime_I_p10"}}