<s>[INST] Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ 
Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value.

Since it is generally hard to find the minimum of a multivariate polynomial, I tried factoring it at first. The $abc$ reminds me of AM-GM, but I wasn't sure how to relate the $a^3+b^3+c^3$ to it. Also the condition that we are working with integers may make it easier to find the minimum. 
 [/INST] The answer is $4$, for $a = 1, b = 1, c = 2$ (or some permutation).
We have 
$$a^3 + b^3 + c^3 - 3abc = (a + b + c)[(a + b + c)^2 - 3(ab + bc + ca)].$$
This shows that any positive value taken by the expression must be at least as large as $a + b + c$, which leaves only the above cases and $a =b = c = 1$ to examine.
Edit Let $A = a^3 - 3abc + b^3 + c^3$. We will write $A$ in terms of the elementary symmetric polynomials. The leading term in lexicographic order is $a^3$, so we compute
$$A - (a + b + c)^3 = - 3a^2b - 3a^2 c - 3ab^2 - 9abc - 3ac^2 - 3b^2 c.$$
The leading term in this expression is $-3a^2b$, so we compute
$$A - (a + b + c)^3 + 3(a + b + c)(ab + ac + bc) = 0.$$ 
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