{"status": "success", "data": {"description_md": "On the coordinate plane, let $A$, $B$, and $C$ be points such that the distance from the origin to each of these points is $25$. Denote $H$ as the orthocenter of $\\triangle ABC$. If the sum of the coordinates of $A$, $B$, and $C$ are $31$, $5$, and $-25$, respectively, find the sum of all possible values of the sum of the coordinates of $H$.", "description_html": "<p>On the coordinate plane, let <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, and <span class=\"katex--inline\">C</span> be points such that the distance from the origin to each of these points is <span class=\"katex--inline\">25</span>. Denote <span class=\"katex--inline\">H</span> as the orthocenter of <span class=\"katex--inline\">\\triangle ABC</span>. If the sum of the coordinates of <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, and <span class=\"katex--inline\">C</span> are <span class=\"katex--inline\">31</span>, <span class=\"katex--inline\">5</span>, and <span class=\"katex--inline\">-25</span>, respectively, find the sum of all possible values of the sum of the coordinates of <span class=\"katex--inline\">H</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Holiday Contest 2025 - Team Round - Problem 17", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}