{"status": "success", "data": {"description_md": "**POTD June 8, 2024**\n\nSupposed $\\triangle ABC$ has side lengths $AB = 14, AC = 13$, and $BC = 15$. Point $D$ is chosen in the interior of $AB$ and point $E$ is selected uniformly at random from $AD$. Point $F$ is then defined to be the intersection point of the perpendicular to $AB$ at $E$ and the union of segments $AC$ and $BC$. Suppose that $D$ is chosen such that the expected value of the length of $EF$ is maximized. Find $AD^2$.", "description_html": "<p><strong>POTD June 8, 2024</strong></p>&#10;<p>Supposed <span class=\"katex--inline\">\\triangle ABC</span> has side lengths <span class=\"katex--inline\">AB = 14, AC = 13</span>, and <span class=\"katex--inline\">BC = 15</span>. Point <span class=\"katex--inline\">D</span> is chosen in the interior of <span class=\"katex--inline\">AB</span> and point <span class=\"katex--inline\">E</span> is selected uniformly at random from <span class=\"katex--inline\">AD</span>. Point <span class=\"katex--inline\">F</span> is then defined to be the intersection point of the perpendicular to <span class=\"katex--inline\">AB</span> at <span class=\"katex--inline\">E</span> and the union of segments <span class=\"katex--inline\">AC</span> and <span class=\"katex--inline\">BC</span>. Suppose that <span class=\"katex--inline\">D</span> is chosen such that the expected value of the length of <span class=\"katex--inline\">EF</span> is maximized. Find <span class=\"katex--inline\">AD^2</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #181", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}