2023 WSMO Accuracy Round Problems/Problem 9

Problem

Given circles $\omega_1, \omega_2$ with radius $1,4$ respectively, they are externally tangent to each other. The diameters of $\omega_1$ , $\omega_2$ are $AB,CD$ respectively, satisfying $AB\parallel CD$ and $BD$ is an external tangent of the circles. The third circle $\omega_3$ passes through $A,C$ and is tangent to $BD$ . If the minimum possible value of the radius of $\omega_3$ is $\frac{a+b\sqrt{c}}{d}$ , where $\gcd(a,b,d) = 1,$ $a$ is positive, and $c$ is squarefree, find $a+b+c+d.$

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2023 WSMO Accuracy Round Problems/Problem 9

Problem

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