{"status": "success", "data": {"description_md": "For real number $x$ let $\\lfloor x\\rfloor$ be the greatest integer less than or equal to $x$, and define $\\{x\\}=x-\\lfloor x\\rfloor$ to be the fractional part of $x$. For example, $\\{3\\}=0$ and $\\{4.56\\}=0.56$. Define $f(x)=x\\{x\\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\\leq x\\leq 2020$. Find the remainder when $N$ is divided by $1000$.\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 real number <span class=\"katex--inline\">x</span> let <span class=\"katex--inline\">\\lfloor x\\rfloor</span> be the greatest integer less than or equal to <span class=\"katex--inline\">x</span>, and define <span class=\"katex--inline\">\\{x\\}=x-\\lfloor x\\rfloor</span> to be the fractional part of <span class=\"katex--inline\">x</span>. For example, <span class=\"katex--inline\">\\{3\\}=0</span> and <span class=\"katex--inline\">\\{4.56\\}=0.56</span>. Define <span class=\"katex--inline\">f(x)=x\\{x\\}</span>, and let <span class=\"katex--inline\">N</span> be the number of real-valued solutions to the equation <span class=\"katex--inline\">f(f(f(x)))=17</span> for <span class=\"katex--inline\">0\\leq x\\leq 2020</span>. Find the remainder when <span class=\"katex--inline\">N</span> is divided by <span class=\"katex--inline\">1000</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": 6, "problem_name": "2020 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/20_aime_II_p15", "prev": "/problem/20_aime_II_p13"}}