The distance covered in races follow a quadratic function. For a race of n
milliseconds, if you hold the buttons for x milliseconds, your boat will go a
distance of (n-x) * x millimeters.
f(x) = (n-x) * x
= -1 * x**2 + n * x
If we are looking to beat a distance d, we look for f(x) > d.
f(x) > d => f(x) - d > 0
=> -1 * x**2 + n * x - d > 0
This is the quadratic function ax^2 + bx + c where:
a is -1
b is n (the time limit for the race)
c is -d (the distance record to beat)
We can find the roots of this quadratic function using the quadratic formula:
-b +/- sqrt(b^2 - 4ac)
----------------------
2a
We need to beat the record, so we can find the two integers that are greater than each roots and count the integers in that range.