{"status": "success", "data": {"description_md": "Bessie the farmer is lining up her $20$ sheep for their daily rations of food. There are a couple of conditions that the line must follow:\n\n- Odd numbered sheep cannot be adjacent to other odd numbered sheep.\n- Even numbered sheep cannot be adjacent to other even numbered sheep.\n- For two sheep numbered $a$ and $b$, if $a = b^n$ for an integer $n > 1,$ then $a$ must be to the right of $b$. For example, $9$ must be to the right of $3$ because $9=3^2$.\n\nIf the number of ways the sheep can be arranged in a line can be expressed as $\\dfrac{(10!)^2}{n},$ where $n$ is a positive integer, find $n$.", "description_html": "<p>Bessie the farmer is lining up her <span class=\"katex--inline\">20</span> sheep for their daily rations of food. There are a couple of conditions that the line must follow:</p>&#10;<ul>&#10;<li>Odd numbered sheep cannot be adjacent to other odd numbered sheep.</li>&#10;<li>Even numbered sheep cannot be adjacent to other even numbered sheep.</li>&#10;<li>For two sheep numbered <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span>, if <span class=\"katex--inline\">a = b^n</span> for an integer <span class=\"katex--inline\">n &gt; 1,</span> then <span class=\"katex--inline\">a</span> must be to the right of <span class=\"katex--inline\">b</span>. For example, <span class=\"katex--inline\">9</span> must be to the right of <span class=\"katex--inline\">3</span> because <span class=\"katex--inline\">9=3^2</span>.</li>&#10;</ul>&#10;<p>If the number of ways the sheep can be arranged in a line can be expressed as <span class=\"katex--inline\">\\dfrac{(10!)^2}{n},</span> where <span class=\"katex--inline\">n</span> is a positive integer, find <span class=\"katex--inline\">n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "Mock Euclid 2025 - Problem 7 Part B", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}