MHB Problem involving arithmetic and geometric mean.

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For three positive numbers \(a, b, c\) such that \(a + b + c = 1\), it is proven that \(ab^2c^3 \leq \frac{1}{432}\). The proof utilizes the arithmetic mean (AM) and geometric mean (GM) inequality by considering the numbers \(a, \frac{b}{2}, \frac{b}{2}, \frac{c}{3}, \frac{c}{3}, \frac{c}{3}\). The AM is calculated to be \(\frac{1}{6}\), while the GM is expressed in terms of \(ab^2c^3\). The inequality \(AM \geq GM\) leads to the conclusion that \(\frac{2^2 3^3}{6^6} \geq ab^2c^3\), confirming the original statement. This establishes a clear relationship between the means and the product of the variables.
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$a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that

$$ab^2c^3 \leq \frac{1}{432}$$
 
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Consider the 6 numbers

$$a,\frac{b}{2},\frac{b}{2},\frac{c}{3},\frac{c}{3},\frac{c}{3}$$

The arithmetic mean of these numbers is

$\displaystyle AM = \dfrac{a+\frac{b}{2}+\frac{b}{2}+\frac{c}{3}+\frac{c}{3}+\frac{c}{3}}{6}$

$=\frac{1}{6}$

Similarly, you can calculate the Geometric Mean.

$\displaystyle GM=\left( \frac{b}{2}\frac{b}{2}\frac{c}{3}\frac{c}{3}\frac{c}{3}\right)^{\frac{1}{6}}=\left( \frac{ab^2 c^3}{2^2 3^3}\right)^{1 \over 6}$

$AM \geq GM$
$\displaystyle \frac{1}{6} \geq \left( \frac{ab^2 c^3}{2^2 3^3}\right)^{1 \over 6}$

$\displaystyle \Rightarrow \frac{2^23^3}{6^6} \geq ab^2c^3$
 
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