Inequality of Four Variables: Prove Σab(a^2+b^2+c^2)≤2

[anemone]

Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$.

Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.

 

[MarkFL]

Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$.

Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.

Please post the solution you have ready. (Time) (Wasntme)

You'd think I would've let you slide on this given that you've posted so many problems, but that's just not how I roll. (Bandit)