Let $a,$ $b,$ $c$ be positive real numbers. Find the minimum value of \[\frac{(a + b)(a + c)(b + c)}{abc}.\]

My strategy that is incorrect:

Expanding the 3 parenthesis, we get:

$(a^2b)/(abc)+(a^2c)/(abc)+(2abc)/(abc)+...$

we then take the arithmetic mean which is that over $7$, and the 7th root of 2 because you take the product

then I get 2 to the 7th root

Guest Feb 4, 2021