{"status": "success", "data": {"description_md": "Let $\\varepsilon$ be the ellipse defined by $16x^2 + 25y^2 = 400$. Let $A$ be the point $(5,0)$ and $B$ be the $(0,-4)$. There is a unique pair of points $(H,K)$, both different from $A$ and $B$, such that the circle with diameter $HK$ passes through both $A$ and $B$. Let $F$ be the point of intersection of the tangents from the points $H$ and $K$ to $\\varepsilon$. Find the sum of the coordinates of $F$.", "description_html": "<p>Let <span class=\"katex--inline\">\\varepsilon</span> be the ellipse defined by <span class=\"katex--inline\">16x^2 + 25y^2 = 400</span>. Let <span class=\"katex--inline\">A</span> be the point <span class=\"katex--inline\">(5,0)</span> and <span class=\"katex--inline\">B</span> be the <span class=\"katex--inline\">(0,-4)</span>. There is a unique pair of points <span class=\"katex--inline\">(H,K)</span>, both different from <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span>, such that the circle with diameter <span class=\"katex--inline\">HK</span> passes through both <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span>. Let <span class=\"katex--inline\">F</span> be the point of intersection of the tangents from the points <span class=\"katex--inline\">H</span> and <span class=\"katex--inline\">K</span> to <span class=\"katex--inline\">\\varepsilon</span>. Find the sum of the coordinates of <span class=\"katex--inline\">F</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Grinding Aces Math Exam 2025 - Problem 12", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}