{"status": "success", "data": {"description_md": "The $ 52$ cards in a deck are numbered $ 1, 2, \\ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a+9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\\ge\\frac12$ can be written as $ \\frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m+n$.\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>The $ 52$ cards in a deck are numbered $ 1, 2, \\ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a+9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\\ge\\frac12$ can be written as $ \\frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m+n$.</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": 6, "problem_name": "2010 AIME II Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/10_aime_II_p14", "prev": "/problem/10_aime_II_p12"}}