Question 14
In this paper, let $[x]$ denote the greatest integer not exceeding x. For examples, $[5]=5$, $[2.8]=2$, and $[-2.3]=-3$.
Assume that $$f=\sqrt{3 a_1+1}+\sqrt{3 a_2+1}+\sqrt{3 a_3+1}+\cdots+\sqrt{3 a_{2016}+1}$$ where $a_1,a_2,\ldots,a_{2016}$ are real numbers such that $a_1+a_2+\cdots+a_{2016}=1$. Find the maximum value of [f].
Assume that $$f=\sqrt{3 a_1+1}+\sqrt{3 a_2+1}+\sqrt{3 a_3+1}+\cdots+\sqrt{3 a_{2016}+1}$$ where $a_1,a_2,\ldots,a_{2016}$ are real numbers such that $a_1+a_2+\cdots+a_{2016}=1$. Find the maximum value of [f].