10. Find all such triples of natural numbers: the product of any two numbers plus 1 is divisible by twice the third number.

Find all such triples of natural numbers: the product of any two numbers plus 1 is divisible by twice the third number.
10. There is only one such 3-tuple $(1,1,1)$. Hint: If the 3-tuple of natural numbers $(a, b, c)$ satisfies the condition, then $a, b, c$ should be coprime and all odd. Since $a b+b c+c a+1$ should be divisible by each of $a, b, c$, it should therefore be divisible by their product $a b c$.