2022 SSMO Relay Round 2 Problems/Problem 1

Problem

Let $P$ be a randomly selected point on a circle, and let $A$ be a randomly selected point inside the same circle. A dilation centered at $P$ with a scale factor of $2$ sends $A$ to $A'.$ Given that the probability that $PA'$ is less than the length of the diameter of the circle can be expressed as $\frac{a\pi+b\sqrt{c}}{d\pi},$ where $a,b,c,d$ are integers such that $a$ and $d$ are positive, $c$ is squarefree, and $\gcd{(a, b, d)}=1$ , find the value of $a+b+c+d.$

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2022 SSMO Relay Round 2 Problems/Problem 1

Problem

Solution