Given positive real numbers𝑥1,𝑥2,…,𝑥2009x1,x2,…,x2009, consider the sum:
𝑥1𝑥1+𝑥2+𝑥2𝑥2+𝑥3+⋯+𝑥2009𝑥2009+𝑥1x1+x2
x1
+x2+x3
x2
+⋯+x2009+x1
x2009
I am trying to determine the least constant𝑀M such that the inequality
𝑥1𝑥1+𝑥2+𝑥2𝑥2+𝑥3+⋯+𝑥2009𝑥2009+𝑥1<𝑀x1+x2
x1
+x2+x3
x2
+⋯+x2009+x1
x2009
<Mholds for all positive real numbers𝑥1,𝑥2,…,𝑥2009x1,x2,…,x2009. How can I find this constant M?
