I was thinking about floor functions and it seemed intuitive to me that for all positive integers $x$ and for any positive integer $w$ where $w+1 \le x$, it follows that:
$$\left\lfloor\frac{(x+1)^2}{w}\right\rfloor - \left\lfloor\frac{x^2}{w}\right\rfloor \ge \left\lfloor\frac{(x+1)^2}{w+1}\right\rfloor - \left\lfloor\frac{x^2}{w+1}\right\rfloor$$
Is my intuition correct? Is there a straight forward argument to establish this? If not, is this an open question that does not have a clear answer?
I was not able to find a counter example. Is there an obvious counter example that I missed?
Computations show that the smallest counterexample is $(x,y)=(17,13)$.
There are $388$ counterexamples with $x\le100$, and $72{,}301$ counterexamples with $x\le1{,}000$. For $x=991$, there are $160$ values of $w$ violating the proposed inequality.
The proportion of $w$s that violate the inequality for a given $x$ seems to be stabilizing around $15\%$.