2025 AMC 10A Problems/Problem 22
Solution 1
Descartes' circle theorem (curvatures 
= 
)
![\[k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1k_2 + k_2k_3 + k_3k_1}.\]](http://latex-new.aopstest.com/a/c/e/ace1fd76350a19ed13018724594c88379a52c55b.png)
For radii 1, 2, 3 we have
![\[k_1 = 1,\quad k_2 = \frac{1}{2},\quad k_3 = \frac{1}{3}.\]](http://latex-new.aopstest.com/f/2/5/f258b45e6cdba2693a50863f7d5336f51b1353a9.png)
Compute the sum and the square-root term
![\[k_1+k_2+k_3 = \frac{11}{6},\qquad k_1k_2+k_2k_3+k_3k_1 = 1.\]](http://latex-new.aopstest.com/1/9/8/198e6057acd88faefc7b97ea8a62520b397f5233.png)
Therefore
![\[k_4 = \frac{11}{6} \pm 2.\]](http://latex-new.aopstest.com/c/b/8/cb83fa1b7268293ff2a7fc9758da66c57b42aa92.png)
Choose the plus sign for the small circle tangent externally to the three given circles
![\[k_4 = \frac{11}{6} + 2 = \frac{23}{6}, \qquad r_4 = \frac{1}{k_4} = \frac{6}{23}.\]](http://latex-new.aopstest.com/7/6/c/76c158cbe172d2ae04d2617e3daba48860cc1acc.png)
~Jonathanmo and another person
