<s>[INST] Maximization of an integer input function Maximize the value of the function 
$$
z=\frac{ab+c}{a+b+c}, 
$$
where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily
distinct from  each other.
Please provide a proof, and if possible a general technique.
Thank you.
 [/INST] $$z=\frac{ab+c}{a+b+c}=\frac{ab-a-b}{a+b+c}+1$$
Clearly, this will be maximum if $c$ minimum $=1$
So, $$z\le \frac{ab-a-b}{a+b+1}+1=\frac{ab+1}{a+b+1} $$
$\frac{a_1b_1+1}{a_1+b_1+1}$ will be greater than $\frac{a_2b_2+1}{a_2+b_2+1}$
if $(a_1a_2-1)(b_1-b_2)+(b_1b_2-1)(a_1-a_2)+a_1b_1-a_2b_2>0$
if $(a_1a_2-1)(b_1-b_2)+(b_1b_2-1)(a_1-a_2)+(a_1-a_2)b_1+a_2(b_1-b_2)>0$
if $(a_1a_2-1+a_2)(b_1-b_2)+(b_1b_2-1+b_1)(a_1-a_2)>0$
Clearly, $\frac{ab+1}{a+b+1} $ increases with the increment of $a,b$ or both.
So, $\frac{ab+1}{a+b+1} $ will be maximum if $a,b$ are maximum.
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