Difference between revisions of "2015 AMC 12A Problems/Problem 25"
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<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath> | <cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath> | ||
| − | + | <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<=5;i=i+1) | ||
| + | { | ||
| + | int skip = trav[i]; | ||
| + | for(int k=skip;k<=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<=63;i=i+1) | ||
| + | { | ||
| + | filldraw(circle(coord[i],coord[i].y),gray(0.75)); | ||
| + | }</asy> | ||
<math> \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}</math> | <math> \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}</math> | ||
Revision as of 02:06, 5 February 2015
Problem
A collection of circles in the upper half-plane, all tangent to the 
-axis, is constructed in layers as follows. Layer 
consists of two circles of radii 
and 
that are externally tangent. For 
, the circles in 
are ordered according to their points of tangency with the -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 
consists of the 
circles constructed in this way. Let 
, and for every circle 
denote by 
its radius. What is
![\[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]](http://latex.artofproblemsolving.com/4/1/e/41e763e89db83d7ae3c444f9f9a448eb3a124b9e.png)
![[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<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=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<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]](http://latex.artofproblemsolving.com/1/1/9/11999bae3eb2df3369375c128acf2c1551a21a07.png)
$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$ (Error compiling LaTeX. Unknown error_msg)
Solution
See Also
| 2015 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 24 |
Followed by Last Problem |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
