{"status": "success", "data": {"description_md": "A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\\ge1$, the circles in $\\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is\n\n$$\\sum_{C\\in S} \\frac{1}{\\sqrt{r(C)}}?$$<br><center><img class=\"problem-image\" alt=\"[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i&lt;=5;i=i+1) { int skip = trav[i]; for(int k=skip;k&lt;=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i&lt;=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]\" class=\"latexcenter\" height=\"262\" src=\"https://latex.artofproblemsolving.com/1/1/9/11999bae3eb2df3369375c128acf2c1551a21a07.png\" width=\"585\"/></center>\n\n$\\textbf{(A)}\\ \\frac{286}{35} \\qquad\\textbf{(B)}\\ \\frac{583}{70} \\qquad\\textbf{(C)}\\ \\frac{715}{73}\\qquad\\textbf{(D)}\\ \\frac{143}{14} \\qquad\\textbf{(E)}\\ \\frac{1573}{146}$\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 collection of circles in the upper half-plane, all tangent to the  <span class=\"katex--inline\">x</span> -axis, is constructed in layers as follows. Layer  <span class=\"katex--inline\">L_0</span>  consists of two circles of radii  <span class=\"katex--inline\">70^2</span>  and  <span class=\"katex--inline\">73^2</span>  that are externally tangent. For  <span class=\"katex--inline\">k\\ge1</span> , the circles in  <span class=\"katex--inline\">\\bigcup_{j=0}^{k-1}L_j</span>  are ordered according to their points of tangency with the  <span class=\"katex--inline\">x</span> -axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer  <span class=\"katex--inline\">L_k</span>  consists of the  <span class=\"katex--inline\">2^{k-1}</span>  circles constructed in this way. Let  <span class=\"katex--inline\">S=\\bigcup_{j=0}^{6}L_j</span> , and for every circle  <span class=\"katex--inline\">C</span>  denote by  <span class=\"katex--inline\">r(C)</span>  its radius. What is</p>&#10;<p> <span class=\"katex--display\">\\sum_{C\\in S} \\frac{1}{\\sqrt{r(C)}}?</span> <br/><center><img class=\"latexcenter\" alt=\"[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i&lt;=5;i=i+1) { int skip = trav[i]; for(int k=skip;k&lt;=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i&lt;=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]\" height=\"262\" src=\"https://latex.artofproblemsolving.com/1/1/9/11999bae3eb2df3369375c128acf2c1551a21a07.png\" width=\"585\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{286}{35} \\qquad\\textbf{(B)}\\ \\frac{583}{70} \\qquad\\textbf{(C)}\\ \\frac{715}{73}\\qquad\\textbf{(D)}\\ \\frac{143}{14} \\qquad\\textbf{(E)}\\ \\frac{1573}{146}</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": "2015 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/15_amc12A_p24"}}