The contest has already concluded more than 6 months ago. I am trying to figure out a smart way to solve the following question below: Let $k$ be an integer. If the equation $$(x-1)|x+1|=x + \frac{k}{2020}$$ has 3 distinct real roots, how many different possible values of $k$ are there?
Here is my initial try: I have split the LHS using test point to get the piece wise function as below: for $x\geq1$: $$ x^2-1 = x + \frac{k}{2020}, $$ for $x<-1$: $$ x^2 -1 = -\left(x+ \frac{k}{2020}\right). $$ Plotting the LHS as upward opening parabola shifted down by 1 units and RHS as 2 straight lines with slope $+1$ and $-1$ with $y$-intercept of $\frac{k}{2020}$ and $-\frac{k}{2020}$. To meet the 3 solutions criteria, the parabola cuts the positively sloped line at 2 points while being tangent to the line at $x=-1/2$ with negative slope.
How should I graphically get to the solution from here?
The function $$f(x)=(x-1)|x+1|-x$$ passes through any horizontal line three times provided that the horizontal line is in the range of $y\in(-\frac54,1)$. Therefore, it suffices to find the number of integers $k$ such that $-\frac54<\frac k{2020}<1.$
Solving the inequality, we get $$-2525<k<2020$$ and we see that $k$ can take on any one of $\boxed{4544}$ integers.