Come on, the information is not at all sufficient!

Algebra Level 5

Consider $k$ positive real numbers: $a_1,a_2,a_3,\ldots , a_{k}$ such that $\displaystyle\prod_{i=1}^{k} a_i=2016$ and $\displaystyle\sum_{i=1}^{k} a_i^{k}=2016 k$ for some positive integer $k$ .

Then, find the value of $\displaystyle\sum_{i=1}^{2014} \dfrac{a_i}{a_{i+1}+a_{i+2}}$ .

The answer is 1007.

1 solution

Nihar Mahajan
Apr 25, 2016

By doing $AM-GM$ inequality, we can easily establish $\displaystyle\sum_{i=1}^{k} a_i^{k}\geq 2016k$ but its is given that it exactly equals $2016k$ , thus we consider the equality case giving $a_1=a_2=a_3=\dots=a_{k}$ . Thus our required sum $\displaystyle\sum_{i=1}^{2014} \dfrac{a_i}{a_{i+1}+a_{i+2}}= \displaystyle\sum_{i=1}^{2014} \dfrac{a_i}{2a_i} = \dfrac{1}{2}\displaystyle\sum_{i=1}^{2014} 1 = \dfrac{2014}{2}=\boxed{1007}$

Moderator note:

Good usage of the equality case of AM-GM.

Come on, the information is not at all sufficient!

The answer is 1007.

1 solution

Moderator note:

0 pending reports