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This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

For positive real numbers \( a, b, c \), find the minimum integer value possible of the following equation:

\[ 6a^{3} + 9b^{3} + 32c^{3} + \frac{1}{4abc} \]

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Find the minimum value of \(x^6 + y^6 -54xy \), where \(x\) and \(y\) are real numbers.

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\( \alpha, \beta > 0 \).

With \( \alpha, \beta > 0 \), if \( \alpha + \dfrac{1}{\alpha}\) and \( 2 - \beta - \dfrac{1}{\beta}\) are the roots of the quadratic equation

\[x^{2} -2(a+1)x + a - 3 = 0,\]

then find the sum of integral values of \(a\).

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Let \(x, y, z\geq 0\) be reals such that \(x+y+z=1\). Find the maximum possible value of

\[x (x+y)^{2}(y+z)^{3}(x+z)^{4}.\]

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