Let be positive real numbers such that . Prove that

Singapore Team Selection Test 2008

Inequalities

Medium

Inequalities by BJ Venkatachala

Do you really need a hint? Try it first!

Use the method of contradiction.

Suppose that . Then we have $latex \Sigma \frac{1}{k(n-1+x_i)}=1$.

Let’s call . We have where . Hence . Using the definition of , this becomes . To get a contradiction, find an inequality that contradicts this one.

Note that . Hence, for any given , . Taking product over all , we get which is a contradiction (given hint 3).

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