Inequality Problem

Here is a very similar solution, except made for all the $\LaTeX$ lovers :) Enjoy!

Note that by AM-GM, we see that $\frac{x + \frac{1}{2015}y + \frac{1}{2015}y + \cdots + \frac{1}{2015}y}{2016} \ge \sqrt[2016]{x \cdot \frac{y}{2015} \cdots \frac{y}{2015}} = \sqrt[2016]{\frac{xy^{2015}}{2015^{2015}}}.$ Now we raise both sides to the $2016$ th power: $\left ( \frac{x + \frac{1}{2015}y + \cdots + \frac{1}{2015}y}{2016} \right)^{2016} = \left( \frac{1}{2016^{2016}} \right) \ge \frac{xy^{2015}}{2015^{2015}}.$ Finally, we isolate $xy^{2015}$ to get our answer: $\frac{2015^{2015}}{2016^{2016}} \ge xy^{2015}.$ So our answer must be $\max{xy^{2015}} = \boxed{\frac{2015^{2015}}{2016^{2016}}}.$

Simple use of AM GM inequality. Split y into 2015 equal parts and apply AM>=GM as both x & y are positive.