<s>[INST] Upper bound for a+b+c in terms of ab+bc+ac I am given a triple of positive integers $a,b,c$ such that $a \geq 1$ and $b,c \geq 2$.
I would like to find an upper bound for $a+b+c$ in terms of $n = ab+bc+ac$. Clearly $a+b+c < ab+bc+ac = n$.
Is there any sharper upper bound that could be obtained (perhaps asimptotically)?
 [/INST] The bound $n/2 + 1$ is tight. First, it is a bound because

$a + b + c \leq a b/2 + b c /2 + a c \leq n/2 + a c / 2 \leq n/2 + 1$

Equality is achieved when $a = 1$ and $b = c = 2$, since then we get $a + b + c = 5$ and $n/2 + 1 = (2+4+2)/2 + 1 = 5$.
Note that there might be other tight bounds that are also functions of $n$.
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