{"status": "success", "data": {"description_md": "For each integer $n \\ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\\le x \\le n$ and $0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor$, where $\\left\\lfloor \\sqrt x \\right\\rfloor$ is the greatest integer not exceeding $\\sqrt x$. Find the number of values of $n$ with $2\\le n \\le 1000$ for which $A(n)$ is an integer.\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 each integer <span class=\"katex--inline\">n \\ge 2</span>, let <span class=\"katex--inline\">A(n)</span> be the area of the region in the coordinate plane defined by the inequalities <span class=\"katex--inline\">1\\le x \\le n</span> and <span class=\"katex--inline\">0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor</span>, where <span class=\"katex--inline\">\\left\\lfloor \\sqrt x \\right\\rfloor</span> is the greatest integer not exceeding <span class=\"katex--inline\">\\sqrt x</span>. Find the number of values of <span class=\"katex--inline\">n</span> with <span class=\"katex--inline\">2\\le n \\le 1000</span> for which <span class=\"katex--inline\">A(n)</span> is an integer.</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": "2015 AIME I Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/15_aime_I_p15", "prev": "/problem/15_aime_I_p13"}}