{"status": "success", "data": {"description_md": "**POTD June 29, 2024**\n\nLet $ABCD$ be the bottom face of a cube, and let $PQRS$ be the top face of that same cube, such that $AP, BQ, CR,$ and $DS$ form vertical edges. Let $\\theta$ be the angle of intersection between space diagonals $AR$ and $BS$. If $\\sin(\\theta)+\\cos(\\theta)+\\tan(\\theta)$ can be expressed as $\\frac{a+b\\sqrt c}{d}$, where $a, b, c, d \\in \\mathbb{Z}^+$, $\\gcd(a, b, d) = 1$, and $c$ is not divisible by the square of any prime, find $1000a+100b+10c+d$.", "description_html": "<p><strong>POTD June 29, 2024</strong></p>&#10;<p>Let <span class=\"katex--inline\">ABCD</span> be the bottom face of a cube, and let <span class=\"katex--inline\">PQRS</span> be the top face of that same cube, such that <span class=\"katex--inline\">AP, BQ, CR,</span> and <span class=\"katex--inline\">DS</span> form vertical edges. Let <span class=\"katex--inline\">\\theta</span> be the angle of intersection between space diagonals <span class=\"katex--inline\">AR</span> and <span class=\"katex--inline\">BS</span>. If <span class=\"katex--inline\">\\sin(\\theta)+\\cos(\\theta)+\\tan(\\theta)</span> can be expressed as <span class=\"katex--inline\">\\frac{a+b\\sqrt c}{d}</span>, where <span class=\"katex--inline\">a, b, c, d \\in \\mathbb{Z}^+</span>, <span class=\"katex--inline\">\\gcd(a, b, d) = 1</span>, and <span class=\"katex--inline\">c</span> is not divisible by the square of any prime, find <span class=\"katex--inline\">1000a+100b+10c+d</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "Problem of the Day #202", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}