{"status": "success", "data": {"description_md": "An $a,b$-lattice grid is defined by the set of all points ($ax+b,ay$) such that $x, y, \\in \\mathbb{Z}$. Consider the $5$,$2$-lattice grid and the $3$,$0$-lattice grid. The probability that a random point picked on the coordinate plane (not necessarily a lattice point) is less than 1 unit away from some point in the $5$,$2$-lattice grid and the $3$,$0$-lattice grid can be expressed as $\\frac{p\\pi-q}{r}$ where $p$, $q$, and $r$ are positive integers with greatest common divisor $1$. Compute $p+q+r$.", "description_html": "<p>An <span class=\"katex--inline\">a,b</span>-lattice grid is defined by the set of all points (<span class=\"katex--inline\">ax+b,ay</span>) such that <span class=\"katex--inline\">x, y, \\in \\mathbb{Z}</span>. Consider the <span class=\"katex--inline\">5</span>,<span class=\"katex--inline\">2</span>-lattice grid and the <span class=\"katex--inline\">3</span>,<span class=\"katex--inline\">0</span>-lattice grid. The probability that a random point picked on the coordinate plane (not necessarily a lattice point) is less than 1 unit away from some point in the <span class=\"katex--inline\">5</span>,<span class=\"katex--inline\">2</span>-lattice grid and the <span class=\"katex--inline\">3</span>,<span class=\"katex--inline\">0</span>-lattice grid can be expressed as <span class=\"katex--inline\">\\frac{p\\pi-q}{r}</span> where <span class=\"katex--inline\">p</span>, <span class=\"katex--inline\">q</span>, and <span class=\"katex--inline\">r</span> are positive integers with greatest common divisor <span class=\"katex--inline\">1</span>. Compute <span class=\"katex--inline\">p+q+r</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Holiday Contest 2025 - Guts Round - Set 5 Problem 1", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}