Editorial for CPC '21 Contest 1 P3 - AQT and Circles

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Author: kevinyang

Subtask 1

For this subtask, it can be observed that the circles $C_2$ ~C_2~ and $C_1$ ~C_1~ will coincide, meaning that the set of possible positions for circle $C_3$ ~C_3~ is equal to the set of its valid positions. This results in the answer always being $1$ ~1~.

Subtask 2

For this subtask, the only case for valid positions that needs to be considered is the case where circle $C_3$ ~C_3~ is located inside of circle $C_1$ ~C_1~ since the probability that circle $C_3$ ~C_3~ lies outside of circle $C_1$ ~C_1~ is negligible. To calculate the probability, you need to calculate the ratio between the area of the set of valid positions and the area of the set of possible positions.
The area of the set of valid positions is $\pi(R_1 - R_3)^2$ ~\pi(R_1 - R_3)^2~ and the area of the set of possible positions is $\pi(R_2 - R_3)^2$ ~\pi(R_2 - R_3)^2~. It can also be observed that $\pi$ ~\pi~ can be cancelled out in the ratio.

Subtask 3

For the full solution, the area of the set of possible positions stays the same but the area of the set of valid positions is not necessarily the same. There are $3$ ~3~ cases to consider:

$C_3$ ~C_3~ is located completely inside of circle $C_1$ ~C_1~
- Condition: $R_3 < R_1$ ~R_3 < R_1~
- Area: $\pi(R_1-R_3)^2$ ~\pi(R_1-R_3)^2~
$C_1$ ~C_1~ is located completely inside of circle $C_3$ ~C_3~
- Condition: $R_3 > R_1$ ~R_3 > R_1~
- Area: $\pi(min(R_3-R_1,R_2-R_3))^2$ ~\pi(min(R_3-R_1,R_2-R_3))^2~
$C_3$ ~C_3~ is located completely outside of circle $C_1$ ~C_1~
- Condition: $R_2-R_1 > 2\cdot R_3$ ~R_2-R_1 > 2\cdot R_3~
- Area: $\pi((R_2-R_3)^2 - (R_1+R_3)^2)$ ~\pi((R_2-R_3)^2 - (R_1+R_3)^2)~

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