2015/round3/boomerang.html

<p>
Today you've found yourself standing on an infinite 2D plane at coordinates 
(<strong>X<sub>0</sub></strong>, <strong>Y<sub>0</sub></strong>). 
There are also <strong>N</strong> targets on this plane, with the <strong>i</strong>th one at coordinates 
(<strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong>).
</p>

<p>
You have a boomerang which you can throw in a straight line in any direction from your initial location. 
After you throw it, you may instantaneously run to any location on the plane. 
After the boomerang has travelled a 
distance of exactly <strong>D</strong> along its initial trajectory, it will return directly to you &mdash; 
that is, to your chosen final location. 
Note that you cannot move around once the boomerang has started its return trip &mdash; 
its path will always consist of 2 line segments (the first of which has a length of exactly <strong>D</strong>). The boomerang and the targets have infinitesimal size.
</p>

<p>
Let <strong>A</strong> be the number of targets which your boomerang hits (directly passes through) 
during the first segment of its flight, and <strong>B</strong> be the number of targets which it hits 
during the second segment. Your throw is then awarded a score of <strong>A</strong> * <strong>B</strong>. 
What's the maximum score you can achieve? Note that, if there is a target at the exact location 
at which the two segments meet (at a distance of <strong>D</strong> from your initial location), 
then it counts towards both <strong>A</strong> and <strong>B</strong>!
</p>


<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of planes.
For each plane, there is first a line containing the space-separated integers 
<strong>X<sub>0</sub></strong> and <strong>Y<sub>0</sub></strong>.
The next line contains the integer <strong>D</strong>, and the one after contains the integer <strong>N</strong>.
Then, <strong>N</strong> lines follow, the <strong>i</strong>th of which contains
the space-separated integers <strong>X<sub>i</sub></strong> and <strong>Y<sub>i</sub></strong>.
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th plane, print a line containing "Case #<strong>i</strong>: " followed by
the maximum score you can achieve.
</p>

<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 20 <br />
1 &le; <strong>N</strong> &le; 3,000 <br />
1 &le; <strong>D</strong> &le; 100 <br />
-100 &le; <strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong> 
&le; 100, for 0 &le; <strong>i</strong> &le; <strong>N</strong> <br />
</p>

<p>
All coordinates are pairwise distinct. The following restrictions are also guaranteed to hold for
the input given:
</p>

<p>
For any three targets at distinct points <strong>a</strong>, <strong>b</strong>, and <strong>c</strong>,
it is guaranteed that <strong>c</strong> is either closer than 10<sup>-13</sup> away from the infinite line
between <strong>a</strong> and <strong>b</strong> (and is considered to be on the line), or is further
than 10<sup>-6</sup> away (and is considered to not be on the line).
</p>

<p>
Let <strong>p</strong> be any point at which the boomerang may change direction after hitting a target.
For any two targets at distinct points <strong>a</strong> and <strong>b</strong>, 
it is guaranteed that <strong>p</strong> is either closer than 10<sup>-13</sup> away from the infinite line
between <strong>a</strong> and <strong>b</strong> (and is considered to be on the line), or is further
than 10<sup>-6</sup> away (and is considered to not be on the line).
</p>


<h3>Explanation of Sample</h3>
<p>
On the first plane, one optimal strategy is to throw the boomerang in the direction of the positive x-axis 
(that is, to (6, 0)), and then run to (0, 0). It will hit targets 2 and 3 on the first segment of its flight, 
and all 3 targets on the second segment, for a score of 2*3=6.
</p>