<s>[INST] Find all prime $a, b, c$ such that $ab+bc+ac > abc$ Find all primes $a, b, c$ such that $ab + bc + ac > abc$
 [/INST] Hint:
The above inequation is equivalent to 
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} > 1$$
where $a,b,c$ are primes 
Complete Solution:

 WLOG we can assume $a \geq b \geq c$ 
 This means that $c<3$ ie. $c=2$ 
 Now, we have $$\frac{1}{a}+\frac{1}{b} > \frac{1}{2}$$ 
 Since $a \geq b$, we have $b<4$ ie $b \in \{2,3\}$ 
 If $b=2$, then $a$ can be any prime number. 
 If $b=3$, then $a<6$, so $a \in \{3,5\}$. 
 So, the required solutions are $(p,2,2),(3,3,2)$ and $(5,3,2)$

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