{"status": "success", "data": {"description_md": "**POTD July 10, 2024**\n\nLet $x, y,$ and $z$ be real numbers such that $x^2+y^2 = 49, y^2+yz+z^2 = 36,$ and $x^2+xz\\sqrt3+z^2 = 25$. If the value of $2xy+yz\\sqrt3+xz$ can be expressed as $a\\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.", "description_html": "<p><strong>POTD July 10, 2024</strong></p>&#10;<p>Let <span class=\"katex--inline\">x, y,</span> and <span class=\"katex--inline\">z</span> be real numbers such that <span class=\"katex--inline\">x^2+y^2 = 49, y^2+yz+z^2 = 36,</span> and <span class=\"katex--inline\">x^2+xz\\sqrt3+z^2 = 25</span>. If the value of <span class=\"katex--inline\">2xy+yz\\sqrt3+xz</span> can be expressed as <span class=\"katex--inline\">a\\sqrt{b}</span>, where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are positive integers and <span class=\"katex--inline\">b</span> is not divisible by the square of any prime, find <span class=\"katex--inline\">a+b</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #213", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}