\( (a+b)(a+c)(b+c)\geq 8abc \)

This inequality, often linked to symmetric polynomials, shows its roots in the theory of inequalities. It reveals that the product of sums of variables can hint at their collective combinations reaching a threshold defined by individual products. A classic application of this is in proving inequalities in competition math or optimizing resource allocation in economics where variables can affect the overall outcome. To tackle this inequality effectively, consider using the AM-GM inequality. First, apply AM-GM to the terms \( (a+b) \), \( (b+c) \), and \( (c+a) \). A common mistake is assuming equality holds effortlessly; remember that AM-GM achieves equality only when all variables involved are equal. Thus, ensuring \( a = b = c \) clarifies the outcome and aids in understanding how these products relate, especially when examining edge cases of variable values.