{"status": "success", "data": {"description_md": "Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$<br><br>$\\includegraphics[width=209, height=124, totalheight=124]{https://latex.artofproblemsolving.com/c/1/8/c1861491cc8a6968ad7722da6de4f99dde44b82b.png}$\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>Square <span class=\"katex--inline\">S_{1}</span> is <span class=\"katex--inline\">1\\times 1.</span> For <span class=\"katex--inline\">i\\ge 1,</span> the lengths of the sides of square <span class=\"katex--inline\">S_{i+1}</span> are half the lengths of the sides of square <span class=\"katex--inline\">S_{i},</span> two adjacent sides of square <span class=\"katex--inline\">S_{i}</span> are perpendicular bisectors of two adjacent sides of square <span class=\"katex--inline\">S_{i+1},</span> and the other two sides of square <span class=\"katex--inline\">S_{i+1},</span> are the perpendicular bisectors of two adjacent sides of square <span class=\"katex--inline\">S_{i+2}.</span> The total area enclosed by at least one of <span class=\"katex--inline\">S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</span> can be written in the form <span class=\"katex--inline\">m/n,</span> where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. Find <span class=\"katex--inline\">m-n.</span><br/><br/><img src=\"https://latex.artofproblemsolving.com/c/1/8/c1861491cc8a6968ad7722da6de4f99dde44b82b.png\" width=\"209\" height=\"124\" class=\"problem-image\"/></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": 3, "problem_name": "1995 AIME Problem 1", "can_next": true, "can_prev": false, "nxt": "/problem/95_aime_p02", "prev": ""}}