For the first ~30\%~, it suffices to consider several cases, since every family of geese moves horizontally. Suppose we check if family ~i~ will be within ~R~ units of Willson's family.

First, check if the family is already within ~R~ units of Willson's family.

Otherwise, suppose Willson's family is moving in one direction. Note that a family can only be within ~R~ units of Willson if their starting ~y~ coordinates are within ~R~ of each other. Suppose the family is heading in the same direction as Willson's family and starts behind Willson. If their speed is faster, then there will eventually be a collision. If their speed is slower, then they will never be close enough. If the family starts ahead of Willson, then there will eventually be a collision if their speed is slower.

Now suppose the family is moving in the opposite direction as Willson. If the family is ahead of Willson, then there will be a collision. Otherwise, there will not.

The second subtask is troll.

For the third subtask, we note that we can find a formula describing the distance between two families with respect to time. If a family moves ~v~ units in ~1~ second in some direction, we can determine the change in ~x~ and change in ~y~ in ~1~ second. In particular, we note that for the ~i^{th}~ family, the ~x~ speed is:

$$\displaystyle \dfrac{v(x_{i2} - x_{i1})}{\sqrt{(x_{i2} - x_{i1})^2 + (y_{i2} - y_{i1})^2}}$$

We can come up with a similar formula for the family's ~y~ speed. Since distance is the product of speed and time, we can then find the exact location of a family after a certain time. Then, for two families, we want a formula that finds the distance between their two locations after a certain time.

We find that this formula is the square root of a quadratic function, so we can easily find the time in which the two families achieve minimum distance. We then check if this minimum distance is less than ~R~.

Do be wary if the family starts within ~R~ units of Willson but moves away, or if the family moves at the same speed and direction as Willson.

Another method of solving the problem involves performing a ternary search on said formula, which passes. A binary search on the "derivative" of the formula however, is extremely inaccurate.

One can also apply knowledge of physics and solve the problem from Willson's frame of reference.

Time Complexity: ~\mathcal{O}(N)~