{"status": "success", "data": {"description_md": "Let $A(0, 0)$ and $B(10, 0)$ be two adjacent vertices of a regular dodecagon ($12$-sided polygon). There exists a third vertex $C \\neq A$ that is adjacent to $B$. The coordinates of $C$ can be either $(a+b\\sqrt{c}, d)$ or $(a+b\\sqrt{c}, e)$, where $a, b, c, d, e$ are integers and $c$ is not divisible by the square of any prime. Find $a + b + c + d + e$.  \n  \n$\\textbf{(A)}~12\\qquad\\textbf{(B)}~28\\qquad\\textbf{(C)}~23\\qquad\\textbf{(D)}~18\\qquad\\textbf{(E)}~13$", "description_html": "<p>Let <span class=\"katex--inline\">A(0, 0)</span> and <span class=\"katex--inline\">B(10, 0)</span> be two adjacent vertices of a regular dodecagon (<span class=\"katex--inline\">12</span>-sided polygon). There exists a third vertex <span class=\"katex--inline\">C \\neq A</span> that is adjacent to <span class=\"katex--inline\">B</span>. The coordinates of <span class=\"katex--inline\">C</span> can be either <span class=\"katex--inline\">(a+b\\sqrt{c}, d)</span> or <span class=\"katex--inline\">(a+b\\sqrt{c}, e)</span>, where <span class=\"katex--inline\">a, b, c, d, e</span> are integers and <span class=\"katex--inline\">c</span> is not divisible by the square of any prime. Find <span class=\"katex--inline\">a + b + c + d + e</span>.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~12\\qquad\\textbf{(B)}~28\\qquad\\textbf{(C)}~23\\qquad\\textbf{(D)}~18\\qquad\\textbf{(E)}~13</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2024 Mock AMC 10 - Problem 10", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}