{"status": "success", "data": {"description_md": "Let $S=\\{0,1,2,3,\\ldots,98,99\\}$. Suppose that Caitlyn is trying to break into Silco's vault, which is protected by two locks, both of which must be open for Caitlyn to enter. The locks each have a passcode that is one of the $100$ elements of $S$, selected independently and at random (meaning that it is possible for the passcodes to be the same). Caitlyn is able to attempt to break into the lock by guessing a random unburned number. If Caitlyn guesses correctly, the lock explodes and is dealt with. Otherwise, that number is burned (so Caitlyn will never guess this number for this lock). Every minute, Caitlyn attempts to break into the vault by trying to break into all intact locks. If the expected number of minutes before Silco's vault is broken into can be written as $\\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a+b$.", "description_html": "<p>Let <span class=\"katex--inline\">S=\\{0,1,2,3,\\ldots,98,99\\}</span>. Suppose that Caitlyn is trying to break into Silco&#8217;s vault, which is protected by two locks, both of which must be open for Caitlyn to enter. The locks each have a passcode that is one of the <span class=\"katex--inline\">100</span> elements of <span class=\"katex--inline\">S</span>, selected independently and at random (meaning that it is possible for the passcodes to be the same). Caitlyn is able to attempt to break into the lock by guessing a random unburned number. If Caitlyn guesses correctly, the lock explodes and is dealt with. Otherwise, that number is burned (so Caitlyn will never guess this number for this lock). Every minute, Caitlyn attempts to break into the vault by trying to break into all intact locks. If the expected number of minutes before Silco&#8217;s vault is broken into can be written as <span class=\"katex--inline\">\\dfrac{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": 6, "problem_name": "Holiday Contest 2025 - Guts Round - Set 7 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}