Mock AIME 1 2010 Problems/Problem 9

Problem

Let $\omega_1$ and $\omega_2$ be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle $\omega_3$ that is internally tangent to both $\omega_1$ and $\omega_2$ , and tangent to the line joining the centers of $\omega_1$ and $\omega_2$ . If the radius of $\omega_3$ can be expressed in the form $a \sqrt{b} - c$ , where $a$ , $b$ , and $c$ are integers, and $b$ is not divisible by the square if any prime, find the value of $a + b + c$ .

Solution

$[asy] import geometry; size(10cm); point A = origin; point B = (10,0); point C = (5-(30-2*sqrt(210)),10*sqrt(210)-144); point D,E,T; circle a = circle(A,5); circle b = circle(B,7); circle c = circle(C,10*sqrt(210)-144); // Defining D, E, and T pair[] d = intersectionpoints(b,A--B); D = d[0]; pair[] e = intersectionpoints(a,A--B); E = e[0]; pair[] t = intersectionpoints(c,A--B); T = t[0]; // Circles draw(a); draw(b); draw(c); // Segments AB, BC, AC, and CT draw(A--B); draw(B--C); draw(A--C); draw(C--T); // Point Labels dot(A); label("A",A,WSW); dot(B); label("B",B,ESE); dot(C); label("C",C,(0,0.5)); dot(D); label("D",D,SW); dot(E); label("E",E,SE); dot(T); label("T",T,S); // Right angle mark markscalefactor=0.05; draw(rightanglemark(C,T,A)); [/asy]$

Mock AIME 1 2010 (Problems, Source)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Mock AIME 1 2010 Problems/Problem 9

Problem

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