Prove that for positive $a,b,c,d$ $$\prod_{cyc}(b+c+d-2a) \le abcd$$
By taking the fourth root, both sides are less than $\frac{a+b+c+d}{4}$.
A common technique used when this happens is to subtract the expression from both sides in the original inequality, do some algebra and hopefully arrive at the result, but I can't seem to find a solution.
It's wrong.
Try $c=d\rightarrow0^+$ and $a=b=1$.