{"status": "success", "data": {"description_md": "**POTD June 4, 2024**\n\nCircles $\\omega_1$ and $\\omega_2$ intersect at points $A$ and $B$. Segment $PQ$ is tangent to $\\omega_1$ at $P$ and $\\omega_2$ at $Q$, and $A$ is closer to $PQ$ than $B$. Point $X$ is on $\\omega_1$ such that $PX || QB$, and point $Y$ is on $\\omega_2$ such that $QY||PB$. Given that $\\angle APQ = 30 ^\\circ$ and $\\angle PQA = 15 ^\\circ$, the ratio $\\frac{AX}{AY}$ can be expressed as $a-\\sqrt b$ for positive integers $a,b$. Find $a+b$.", "description_html": "<p><strong>POTD June 4, 2024</strong></p>&#10;<p>Circles <span class=\"katex--inline\">\\omega_1</span> and <span class=\"katex--inline\">\\omega_2</span> intersect at points <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span>. Segment <span class=\"katex--inline\">PQ</span> is tangent to <span class=\"katex--inline\">\\omega_1</span> at <span class=\"katex--inline\">P</span> and <span class=\"katex--inline\">\\omega_2</span> at <span class=\"katex--inline\">Q</span>, and <span class=\"katex--inline\">A</span> is closer to <span class=\"katex--inline\">PQ</span> than <span class=\"katex--inline\">B</span>. Point <span class=\"katex--inline\">X</span> is on <span class=\"katex--inline\">\\omega_1</span> such that <span class=\"katex--inline\">PX || QB</span>, and point <span class=\"katex--inline\">Y</span> is on <span class=\"katex--inline\">\\omega_2</span> such that <span class=\"katex--inline\">QY||PB</span>. Given that <span class=\"katex--inline\">\\angle APQ = 30 ^\\circ</span> and <span class=\"katex--inline\">\\angle PQA = 15 ^\\circ</span>, the ratio <span class=\"katex--inline\">\\frac{AX}{AY}</span> can be expressed as <span class=\"katex--inline\">a-\\sqrt b</span> for positive integers <span class=\"katex--inline\">a,b</span>. Find <span class=\"katex--inline\">a+b</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Problem of the Day #177", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}