{"status": "success", "data": {"description_md": "Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\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>Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability <span class=\"katex--inline\">\\frac23</span>. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability <span class=\"katex--inline\">\\frac34</span>. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is <span class=\"katex--inline\">\\frac{p}{q}</span>, where <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">q</span> are relatively prime positive integers. Find <span class=\"katex--inline\">p+q</span>.</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": 3, "problem_name": "2022 AIME II Problem 2", "can_next": true, "can_prev": true, "nxt": "/problem/22_aime_II_p03", "prev": "/problem/22_aime_II_p01"}}