AMC 10A 2025 (Problem 22)A circle of radius rrr is surrounded by three circles, whose radii are 111, 222, and 333, all externally tangent to the inner circle and externally tangent to each other, as shown below. What is rrr?(A) 14\text{(A)}\;\frac{1}{4}(A)41(B) 623\text{(B)}\;\frac{6}{23}(B)236(C) 311\text{(C)}\;\frac{3}{11}(C)113(D) 517\text{(D)}\;\frac{5}{17}(D)175(E) 310\text{(E)}\;\frac{3}{10}(E)103Related TopicsCoreCoordinate Geometry FoundationsMajorCircle TangencyMinorEquation Solving TechniquesQuadratic FunctionsCheck AnswerYour answer:ABCDECheckHints (7)Hint 1Hint 2Use coordinate geometry.Hint 3Let O1(0,0)O_1(0,0)O1(0,0), O2(3,0)O_2(3,0)O2(3,0), O3(4,0)O_3(4,0)O3(4,0), and O(x,y)O(x,y)O(x,y).OO12=x2+y2=(1+r)2=1+r2+2r(1)OO_1^2=x^2+y^2=(1+r)^2=1+r^2+2r \qquad (1)OO12=x2+y2=(1+r)2=1+r2+2r(1)OO22=x2+(y−3)2=(2+r)2=4+r2+4r(2)OO_2^2=x^2+(y-3)^2=(2+r)^2=4+r^2+4r \qquad (2)OO22=x2+(y−3)2=(2+r)2=4+r2+4r(2)OO32=(x−4)2+y2=(3+r)2=9+r2+6r(3)OO_3^2=(x-4)^2+y^2=(3+r)^2=9+r^2+6r \qquad (3)OO32=(x−4)2+y2=(3+r)2=9+r2+6r(3)Hint 4From (1) and (2):(y−3)2−y2=2r+3(y-3)^2-y^2=2r+3(y−3)2−y2=2r+39−6y=2r+39-6y=2r+39−6y=2r+3y=3−r3(4)y=\frac{3-r}{3} \qquad (4)y=33−r(4)Hint 5From (1) and (3):(x−4)2−x2=4r+8(x-4)^2-x^2=4r+8(x−4)2−x2=4r+816−8x=4r+816-8x=4r+816−8x=4r+8x=2−r2(5)x=\frac{2-r}{2} \qquad (5)x=22−r(5)Hint 6From (1), (4), and (5):(3−r3)2+(2−r2)2=1+r2+2r\left(\frac{3-r}{3}\right)^2+\left(\frac{2-r}{2}\right)^2=1+r^2+2r(33−r)2+(22−r)2=1+r2+2r⇒4(3−r)2+9(2−r)2=36(1+r2+2r)\Rightarrow 4(3-r)^2+9(2-r)^2=36(1+r^2+2r)⇒4(3−r)2+9(2−r)2=36(1+r2+2r)⇒4(9−6r+r2)+9(4−4r+r2)=36(1+r2+2r)\Rightarrow 4(9-6r+r^2)+9(4-4r+r^2)=36(1+r^2+2r)⇒4(9−6r+r2)+9(4−4r+r2)=36(1+r2+2r)⇒23r2+132r−36=0\Rightarrow 23r^2+132r-36=0⇒23r2+132r−36=0Hint 7Use Quadratic Formula:r=−132±1322−4(23)(−36)46r=\frac{-132\pm\sqrt{132^2-4(23)(-36)}}{46}r=46−132±1322−4(23)(−36)=−66±14423=\frac{-66\pm\sqrt{144}}{23}=23−66±144=−66+12123=\frac{-66+12\sqrt{1}}{23}=23−66+121=623=\frac{6}{23}=236Final Answer(B) 623\frac{6}{23}236Related Problems (13)AMC 10A 2025 (Problem 20)AMC 12A 2025 (Problem 11)AMC 8 2026 (Problem 13)AIME I 2026 (Problem 10)AIME I 2026 (Problem 12)AMC 8 2026 (Problem 23)AMC 10A 2025 (Problem 10)AMC 12A 2025 (Problem 9)AMC 12A 2025 (Problem 24)AMC 10B 2025 (Problem 12)AMC 10B 2025 (Problem 20)AIME I 2025 (Problem 6)AIME I 2026 (Problem 3)View all related problems →
