{"status": "success", "data": {"description_md": "Alice and Bob play in an infinitely long tetris match. The match is consisted of infinitely many games of tetris, each of which is won by either Alice or Bob, but not both. The probability that Alice or Bob wins is constant given the number of times Alice and Bob have won so far, but variable otherwise. For example, if Alice has won $4$ matches and Bob has won $3$, the probability that Alice wins will be the same regardless of how they reached this point. However, this probability may be different if Alice has won $3$ matches and Bob has won $4$.\n\nThese probabilities are selected such that the probability that after $n$ games, the probability that Alice has won $a$ out of the first $n$ games is equal to the probability that she has won $b$ out of the first $n$ games, for all $0 \\le a < b \\le n$. The current score is Alice with $4$ wins and Bob with $3$ wins. The probability that after five more games, Alice and Bob will have an equal number of wins can be written as $a/b$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.", "description_html": "<p>Alice and Bob play in an infinitely long tetris match. The match is consisted of infinitely many games of tetris, each of which is won by either Alice or Bob, but not both. The probability that Alice or Bob wins is constant given the number of times Alice and Bob have won so far, but variable otherwise. For example, if Alice has won <span class=\"katex--inline\">4</span> matches and Bob has won <span class=\"katex--inline\">3</span>, the probability that Alice wins will be the same regardless of how they reached this point. However, this probability may be different if Alice has won <span class=\"katex--inline\">3</span> matches and Bob has won <span class=\"katex--inline\">4</span>.</p>&#10;<p>These probabilities are selected such that the probability that after <span class=\"katex--inline\">n</span> games, the probability that Alice has won <span class=\"katex--inline\">a</span> out of the first <span class=\"katex--inline\">n</span> games is equal to the probability that she has won <span class=\"katex--inline\">b</span> out of the first <span class=\"katex--inline\">n</span> games, for all <span class=\"katex--inline\">0 \\le a &lt; b \\le n</span>. The current score is Alice with <span class=\"katex--inline\">4</span> wins and Bob with <span class=\"katex--inline\">3</span> wins. The probability that after five more games, Alice and Bob will have an equal number of wins can be written as <span class=\"katex--inline\">a/b</span> where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are relatively prime positive integers. Find <span class=\"katex--inline\">a + b</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Grinding Aces Math Exam 2025 - Problem 8", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}