Define $$f(x,a) := (2x-a)\lfloor\frac{x}{a}\rfloor-a\lfloor\frac{x}{a}\rfloor^2.$$ It seems that $$f(x,a)+f(x,b)\geq 2f(x,c),\forall a,b \in [1,x],a+b=2c.$$

I have written a program that has checked this equality for $x \leq 1000$

however, I have struggled with this inequality for many days and found no research papers involving multivariate inequality of floor function ( only basic properties )

Any help is appreciated