Inequality in 5 variables

Question

Let 5 positive real variables $(a,b,c,d,e)$. Prove or disprove: $$ \sum_{cyc} a^2 b d (c+e)\ge \sum_{cyc} a b c e (a+d) $$ where $\sum_{cyc}$ means all 5 cyclic shifts $(a,b,c,d,e) \to (b,c,d,e,a) \to$ etc. Equality occurs if all 5 variables are equal, and it appears that equality occurs at no other points. I couldn't find counterexamples through simulations.

Answer 1

We need to prove that $$\sum_{cyc}a^2b(cd+de-ce)\geq5abcde.$$ We can take $a\rightarrow+\infty$ and $cd+de-ce<0.$

For example, take $(a,b,c,d,e)=(100,1,2,1,3).$

Now, we see that this inequality is wrong.

Answer 2

I need to check the arithmetic, but try $(a,b,c,d,e)=(2,3,1,2,10)$.

LHS$=2036$ and RHS$=2096$.