2016/round3/boomerang_umbrella.html

<p>
Uh oh, the weather forecast predicts that a rainstorm will soon break out over the infinite, 1-dimensional number line you call home!
</p>

<p>
The forecast is remarkably precise. In fact, it's known that exactly <strong>N</strong> raindrops will fall, with the <strong>i</strong>th drop striking the number line at position 
<strong>X<sub>i</sub></strong> exactly <strong>K<sub>i</sub></strong> seconds after the start of the storm.
No two drops will strike the number line at exactly the same position and time.
</p>

<p>
You'd like to stop as many of the drops as possible, but you don't exactly have an umbrella... so your boomerang will have to do. Your plan is as follows:
</p>

<p>
<ol>
<li> Stand at some (possibly non-integral) position A on the number line, and choose some other position B (A &ne; B). </li>
<li> At some point in time, throw your boomerang from A to B. This point in time can be arbitrarily long before or after the start of the storm (the storm won't start for a while, so you have time to prepare). 
        The boomerang will travel along your chosen line segment at a constant speed of S units per second. </li>
<li> Make your boomerang spin in place at position B for some non-negative amount of time (which can be arbitrarily large). </li>
<li> Have your boomerang travel back along the line segment from B to A at the same speed, and catch it as soon as it gets back to position A. </li>
</ol>
</p>

<p>
During the inclusive time interval from when you throw the boomerang to when you catch it again, if the boomerang is ever at exactly the same position as a raindrop at exactly the time that it strikes the number line, 
it will intercept it... just like an umbrella! What's the maximum number of raindrops you can intercept in this way with a single throw?
</p>


<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of rainstorms.
For each rainstorm, there is first a line containing the space-separated integers <strong>N</strong> and <strong>S</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>K<sub>i</sub></strong>.
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th rainstorm, print a line containing "Case #<strong>i</strong>: " followed by
the maximum number of raindrops you can stop.
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 50 <br />
1 &le; <strong>N</strong> &le; 100,000 <br />
1 &le; <strong>S</strong> &le; 1,000,000,000 <br />
0 &le; <strong>X<sub>i</sub></strong>, <strong>K<sub>i</sub></strong> &le; 1,000,000,000 <br>
</p>


<h3>Explanation of Sample</h3>
<p>
In the first rainstorm, one solution is to stand at X = 99 and let your boomerang spin at X = 100 until it intercepts all three raindrops that will fall at that position.

In the second rainstorm, one solution is to stand at X = 50 and throw your boomerang towards X = 100 as soon as the first raindrop hits it. 
If you recall your boomerang as soon as it reaches X = 100, it will hit the second raindrop and be back just in time to hit the third.
</p>