{"status": "success", "data": {"description_md": "A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \\le i \\le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \\le c_2 \\le \\cdots \\le c_{14}$. What is the ratio $\\frac{c_{14}}{c_1}$ for the least possible value of $r$?\n\n$\\textbf{(A)} ~\\frac{21}{5} \\qquad\\textbf{(B)} ~\\frac{85}{13} \\qquad\\textbf{(C)} ~7 \\qquad\\textbf{(D)} ~\\frac{39}{5} \\qquad\\textbf{(E)} ~17$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A circle with integer radius  <span class=\"katex--inline\">r</span>  is centered at  <span class=\"katex--inline\">(r, r)</span> . Distinct line segments of length  <span class=\"katex--inline\">c_i</span>  connect points  <span class=\"katex--inline\">(0, a_i)</span>  to  <span class=\"katex--inline\">(b_i, 0)</span>  for  <span class=\"katex--inline\">1 \\le i \\le 14</span>  and are tangent to the circle, where  <span class=\"katex--inline\">a_i</span> ,  <span class=\"katex--inline\">b_i</span> , and  <span class=\"katex--inline\">c_i</span>  are all positive integers and  <span class=\"katex--inline\">c_1 \\le c_2 \\le \\cdots \\le c_{14}</span> . What is the ratio  <span class=\"katex--inline\">\\frac{c_{14}}{c_1}</span>  for the least possible value of  <span class=\"katex--inline\">r</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} ~\\frac{21}{5} \\qquad\\textbf{(B)} ~\\frac{85}{13} \\qquad\\textbf{(C)} ~7 \\qquad\\textbf{(D)} ~\\frac{39}{5} \\qquad\\textbf{(E)} ~17</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2022 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/22_amc12A_p24"}}