If we start by evaluating ~f(1)~, then ~f(2)~, and so on, we note that ~f~ might be decreasing but eventually starts increasing. We note that ~MX~ increases by ~X~ every time ~M~ increases by ~1~, but eventually ~\dfrac{KP}{M}~ will not decrease quickly enough to offset.

The intended solution is therefore to increase ~X~ until ~f(X) \le f(X+1)~ and report ~f(X)~.

The most common pitfall observed in solutions was contestants not following instructions and printing out the value to an incorrect number of decimal places.

It is possible to use calculus or other techniques to directly evaluate the minimum, but those were not necessary to solve this problem.