TITLE: Inequality Of Four Variables
QUESTION [7 upvotes]: Suppose $a,b,c,d$ are real numbers greater than $1$. Prove that
$$abc+abd+acd+bcd-3abcd<1.$$

REPLY [11 votes]: Let $a=1+x$, $ b=1+y$, $ c=1+z$ and $d=1+w$. Then $abc+acd+abd+bcd-3abcd=1-(xy+xz+xw+yz+yw+zw)-2(xyz+xzw+xyw+yzw)-3xyzw$