{"status": "success", "data": {"description_md": "All $705$ members in the TopsOJ discord server decided to go to a party! They went to a mansion (generously donated by Leading Aces Academy) with $100$ rooms, and each person went to a random room. We label the rooms $1,2,\\ldots, 100$ (where room $i$ has $x_i$ people) such that if $i < j$, then $x_i \\le x_j$. Every minute, some member in room $i$ moves to a room with at least $x_i$ people (not including the member who just moved). Then, the rooms are relabeled as described above. Now, at first, room $i$ has $a_i$ people in it. After $K$ minutes, room $i$ has $b_i$ people in it. If $K$ is maximal, find $$\\sum_{i=1}^{100}|a_i - b_i|.$$", "description_html": "<p>All <span class=\"katex--inline\">705</span> members in the TopsOJ discord server decided to go to a party! They went to a mansion (generously donated by Leading Aces Academy) with <span class=\"katex--inline\">100</span> rooms, and each person went to a random room. We label the rooms <span class=\"katex--inline\">1,2,\\ldots, 100</span> (where room <span class=\"katex--inline\">i</span> has <span class=\"katex--inline\">x_i</span> people) such that if <span class=\"katex--inline\">i &lt; j</span>, then <span class=\"katex--inline\">x_i \\le x_j</span>. Every minute, some member in room <span class=\"katex--inline\">i</span> moves to a room with at least <span class=\"katex--inline\">x_i</span> people (not including the member who just moved). Then, the rooms are relabeled as described above. Now, at first, room <span class=\"katex--inline\">i</span> has <span class=\"katex--inline\">a_i</span> people in it. After <span class=\"katex--inline\">K</span> minutes, room <span class=\"katex--inline\">i</span> has <span class=\"katex--inline\">b_i</span> people in it. If <span class=\"katex--inline\">K</span> is maximal, find <span class=\"katex--display\">\\sum_{i=1}^{100}|a_i - b_i|.</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Holiday Contest 2025 - Guts Round - Set 8 Problem 1", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}