<s>[INST] Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$ Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$.
(Source : Singapore Math Olympiad 2012, Senior section, Round 1, question 29).
I tried using the fact that $2x +13, 8z + 12\ge 0$ to deduce that $y \le 11$, but I couldn't continue from there on. The answer should be an integer, since only integer answers were allowed.
The competition was designed for 15/16 year olds. A simple yet elegant solution would be nice.

For reference of the original problem, 
 [/INST] If $y$ is allowed to be negative, then there is no maximum.
Take $ x =k - \frac{13}{2}$, $ y = -k - \frac{5}{3}$, $ z= 3 + \frac{13}{2} + \frac{5}{3}$
The important part of the value looks like $ \sqrt{2k} + \sqrt[3]{-3k} + C$, which tends to infinity as $k\rightarrow \infty$.

Otherwise, if $y$ is positive, see Math110's solution.
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