parmenides51 wrote: Let $a,b,c,d \in R^+$ with $abcd=1$. Prove that $$\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4$$(dektep) By Bernoulli, it suffices to prove that $$\frac{1+ab}{1+a}+\frac{1+bc}{1+b}+\frac{1+cd}{1+c}+\frac{1+da}{1+d} \geq 4.$$ It is a known result. See e.g. https://artofproblemsolving.com/community/c6h85389p496957 https://artofproblemsolving.com/community/c6h2010928p14094513