{"status": "success", "data": {"description_md": "In the figure below, semicircles with centers at $A$ and $B$ and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $JK$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?<br><center><img class=\"problem-image\" alt=\"[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (-1,0)+(2+6/7)*dir(36.86989); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot(P); [/asy]\" class=\"latexcenter\" height=\"122\" src=\"https://latex.artofproblemsolving.com/c/7/0/c70afdd05a6954478364e08a7b13089dc5022e4a.png\" width=\"238\"/></center>\n\n$\\textbf{(A)}\\ \\frac{3}{4}<br>\\qquad \\textbf{(B)}\\ \\frac{6}{7}<br>\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{3}}{2}<br>\\qquad\\textbf{(D)}\\ \\frac{5}{8}\\sqrt{2}<br>\\qquad\\textbf{(E)}\\ \\frac{11}{12}$\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>In the figure below, semicircles with centers at  <span class=\"katex--inline\">A</span>  and  <span class=\"katex--inline\">B</span>  and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter  <span class=\"katex--inline\">JK</span> . The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at  <span class=\"katex--inline\">P</span>  is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at  <span class=\"katex--inline\">P</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (-1,0)+(2+6/7)*dir(36.86989); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot(P); [/asy]\" height=\"122\" src=\"https://latex.artofproblemsolving.com/c/7/0/c70afdd05a6954478364e08a7b13089dc5022e4a.png\" width=\"238\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{3}{4}\\qquad \\textbf{(B)}\\ \\frac{6}{7}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{3}}{2}\\qquad\\textbf{(D)}\\ \\frac{5}{8}\\sqrt{2}\\qquad\\textbf{(E)}\\ \\frac{11}{12}</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": 3, "problem_name": "2017 AMC 12A Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/17_amc12A_p17", "prev": "/problem/17_amc12A_p15"}}