How to solve these inequalities? How to solve these inequalities?

  
*
  
*If $a,b,c,d \gt 1$, prove that $8(abcd + 1) \gt
 (a+1)(b+1)(c+1)(d+1)$.
  
*Prove that $ \cfrac{(a+b)xy}{ay+bx} \lt \cfrac{ax+by}{a+b}$
  
*Find the greatest value of $x^3y^5z^7$ when $2x^2+2y^2+2x^2=15$
  

Any hints/solution are welcome.

Let $a=1+x$, $b=1+y$, $c=1+z$ and $d=1+t$. Hence,
$$8(abcd + 1)-(a+1)(b+1)(c+1)(d+1)=$$
$$=8(1+x)(1+y)(1+z)(1+t)+8-(2+x)(2+y)(2+z)(2+t)=$$
$$=4(xy+xz+yz+xt+yt+zt)+6(xyz+xyt+xzt+yzt)+7xyzt>0$$
Done!
