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.