{"status": "success", "data": {"description_md": "A $7$-meter long graham cracker is being eaten by Aqua and Blue. Aqua and Blue take turns, starting with Aqua, eating the graham cracker in the following way:\n\n- On each of Aqua's turns, there is a $\\tfrac12$ chance that they eat one meter of the graham cracker, and a $\\tfrac12$ chance they do not take a bite.\n- On each of Blue's turns, there is a $\\tfrac12$ chance, independent of Aqua, that Blue eats two meters or the entire cracker, whichever is smaller, and a $\\tfrac12$ chance they do not take a bite.\n\nIf the probability that Aqua finishes eating the cracker can be expressed as $\\frac{a}{b}$, where $a$ and $b$ are relatively prime, positive integers, find the remainder when $a+b$ is divided by $1000$.\n", "description_html": "<p>A <span class=\"katex--inline\">7</span>-meter long graham cracker is being eaten by Aqua and Blue. Aqua and Blue take turns, starting with Aqua, eating the graham cracker in the following way:</p>&#10;<ul>&#10;<li>On each of Aqua&#8217;s turns, there is a <span class=\"katex--inline\">\\tfrac12</span> chance that they eat one meter of the graham cracker, and a <span class=\"katex--inline\">\\tfrac12</span> chance they do not take a bite.</li>&#10;<li>On each of Blue&#8217;s turns, there is a <span class=\"katex--inline\">\\tfrac12</span> chance, independent of Aqua, that Blue eats two meters or the entire cracker, whichever is smaller, and a <span class=\"katex--inline\">\\tfrac12</span> chance they do not take a bite.</li>&#10;</ul>&#10;<p>If the probability that Aqua finishes eating the cracker can be expressed as <span class=\"katex--inline\">\\frac{a}{b}</span>, where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are relatively prime, positive integers, find the remainder when <span class=\"katex--inline\">a+b</span> is divided by <span class=\"katex--inline\">1000</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "March Break 2024 - Problem 12", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}