{"status": "success", "data": {"description_md": "Powder and Vi are running at constant but different speeds along a figure-eight racetrack, where one road is directly above the other at the center, and which is vertically and horizontally symmetric. Vi is a faster runner than Powder. They both start at the westmost point of the track (red dot), and run in the same or opposite direction. They cannot jump from the higher to the lower road and must follow the white line.\n\n<center><img width=\"300px\" class=\"latex-img\" src=\"/static/christmas2_team_p19.png\"></center>\n\nWe say the two sisters meet at the center for the first time if and only if each is directly on top of the blue central point without having passed each other before. That is, both can meet on the same road, or one can be directly above the other on the opposite road.\n\nGiven that they meet at the center for the first time, the sum of all possible values of $\\left(1-\\frac{\\text{speed of powder}}{\\text{speed of vi}}\\right)^2$ can be written as $\\frac{a}{b}\\pi^2-\\frac{c}{d}$ for relatively prime positive integers $a$ and $b$ and relatively prime positive integers $c$ and $d$. Compute $a+b+c+d$.", "description_html": "<p>Powder and Vi are running at constant but different speeds along a figure-eight racetrack, where one road is directly above the other at the center, and which is vertically and horizontally symmetric. Vi is a faster runner than Powder. They both start at the westmost point of the track (red dot), and run in the same or opposite direction. They cannot jump from the higher to the lower road and must follow the white line.</p>&#10;<center><img width=\"300px\" class=\"latex-img\" src=\"/static/christmas2_team_p19.png\"/></center>&#10;<p>We say the two sisters meet at the center for the first time if and only if each is directly on top of the blue central point without having passed each other before. That is, both can meet on the same road, or one can be directly above the other on the opposite road.</p>&#10;<p>Given that they meet at the center for the first time, the sum of all possible values of <span class=\"katex--inline\">\\left(1-\\frac{\\text{speed of powder}}{\\text{speed of vi}}\\right)^2</span> can be written as <span class=\"katex--inline\">\\frac{a}{b}\\pi^2-\\frac{c}{d}</span> for relatively prime positive integers <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> and relatively prime positive integers <span class=\"katex--inline\">c</span> and <span class=\"katex--inline\">d</span>. Compute <span class=\"katex--inline\">a+b+c+d</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Holiday Contest 2025 - Team Round - Problem 19", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}