Let $a,b,c,d,e,f,g,h$ be 8 non negative real numbers such that $a+b+c+d+e+f+g+h =16$. If $P= ab+bc+cd+de+ef+fg+gh$ then find the maximum value of $P$.
I tried using rearrangement inequality but couldn't get the correct answer.
2026-03-27 01:13:26.1774574006
Inequalities in mathematics
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By AM-GM $$P\leq(a+c+e+g)(b+d+f+h)\leq\left(\frac{a+b+c+d+e+f+g+h}{2}\right)^2=64.$$ The equality occurs for $a=b=8$ and $c=d=e=f=g=h=0,$ which says that we got a maximal value.