A Sample problem from my book

One of the most beautiful inequalities in this stuff is

Let $a,b,c,d$ be non-negative real numbers such that $a^2+b^2+c^2+d^2=1$. Prove that

$a+b+c+d\geq a^3+b^3+c^3+d^3+ab+bc+cd+da+ac+bd$

For other, see

sample-ineqs1.pdf