Question #41858

This problem tests the interaction between the fractional part operator, modulus functions, and definite integrals with variable limits.

The core trap lies in the periodic nature of the integral of a centered fractional part function.

Students often fail to recognize that the integral evaluates to zero over a full period, leading them to incorrectly attempt a direct piecewise integration over the entire domain $[0, x]$, which becomes algebraically intractable.

Furthermore, the problem requires establishing a system of inequalities via the discriminant of the resulting quadratic equation to bound the integer part of $x$.

A secondary 'lethal collision' occurs if students overlook the odd symmetry of the integral function, causing them to miss the negative solutions entirely or miscount the trivial solution at the origin.