{"status": "success", "data": {"description_md": "A rectangle with perimeter $44$ is cut out from a square sheet of paper with side length $a$, such that the sides of the rectangle are parallel to the sides of the square and a vertex of the rectangle lies on top of a vertex of the square. This leaves a concave hexagon such that all interior angles are equal to $90^\\circ$ or $270^\\circ$, and the side length of the square is equal to the area of the hexagon. If the largest value of $a$ can be expressed as $\\frac{p\\sqrt{q} + r}{s}$, where $p, q, r,s$ are all positive integers, $q$ is not divisible by the square of any prime, and $\\gcd(p,r,s)=1$, compute $p + q + r+s$.", "description_html": "<p>A rectangle with perimeter <span class=\"katex--inline\">44</span> is cut out from a square sheet of paper with side length <span class=\"katex--inline\">a</span>, such that the sides of the rectangle are parallel to the sides of the square and a vertex of the rectangle lies on top of a vertex of the square. This leaves a concave hexagon such that all interior angles are equal to <span class=\"katex--inline\">90^\\circ</span> or <span class=\"katex--inline\">270^\\circ</span>, and the side length of the square is equal to the area of the hexagon. If the largest value of <span class=\"katex--inline\">a</span> can be expressed as <span class=\"katex--inline\">\\frac{p\\sqrt{q} + r}{s}</span>, where <span class=\"katex--inline\">p, q, r,s</span> are all positive integers, <span class=\"katex--inline\">q</span> is not divisible by the square of any prime, and <span class=\"katex--inline\">\\gcd(p,r,s)=1</span>, compute <span class=\"katex--inline\">p + q + r+s</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "Holiday Contest 2025 - Guts Round - Set 4 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}