<s>[INST] Maximum Value of $a+b+c$ 
Given that  $a+\frac {11}b+\frac c4=20$, what is the maximum value of $a+b+c$? Here, $a,b,c$ are positive integers.

I tried to find the maximum from the AM-GM relationship but failed.
 [/INST] Notice 
$$b > 0 \implies \frac{11}{b} > 0
\implies a + \frac{c}{4} < 20 \iff 4a+c < 80 \implies 4a + c \le 79$$
This leads to
$$b = \frac{11}{20- \left(a + \frac{c}{4}\right)} = \frac{44}{80 - (4a + c)} \le \frac{44}{80-79} = 44$$
Furthermore, $a \ge 1$ implies
$$a + c \le 79 - 3a \le 79 - 3 = 76$$
Combine these, we can conclude $a + b + c \le 76 + 44 = 120$.
Since this value $120$ is achieved by $(a,b,c) = (1,44,75)$ and
$1 + \frac{11}{44} + \frac{75}{4} = 20$. The maximum value of $a + b + c$ equals to $120$.
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