Question

Sol. [Key Concept: Positive-integer relation ⇒ constrain the values.]

$$\frac{\underset{\text{a times}\text{~}}{\overset{(a + a + \ldots a)}{︸}} + \underset{b\text{~}\text{times}\text{~}}{\overset{(b + b + \ldots b)}{︸}} + \underset{c\text{~}\text{times}\text{~}}{\overset{(c + c + \ldots c)}{︸}}}{a + b + c} \geq \left( a^{a}b^{b}c^{c} \right)^{\frac{1}{a + b + c}}$$

$$\begin{matrix} \Rightarrow & \left( a^{a}b^{b}c^{c} \right)^{1/n} \leq \frac{a^{2} + b^{2} + c^{2}}{n} \\ & \underset{\text{b times}\text{~}}{\overset{(a + a + \ldots a)}{︸}} + \underset{\text{c times}\text{~}}{\overset{(b + b + \ldots b)}{︸}} + \underset{\text{a times}\text{~}}{\overset{(c + c + \ldots c)}{︸}} \geq \left( a^{b}b^{c}c^{a} \right)^{\frac{1}{a + b + c}} \\ & \frac{a^{2} + b^{2} + c^{2}}{n} \geq \frac{ab + bc + ca}{n} \geq \left( a^{b}b^{c}c^{a} \right)^{\frac{1}{n}} \\ & \left( a^{a}b^{b}c^{c} \right)^{\frac{1}{n}} + \left( a^{b}b^{c}c^{a} \right)^{\frac{1}{n}} + \left( a^{c}b^{a}c^{b} \right)^{\frac{1}{n}} \leq \frac{a^{2} + b^{2} + c^{2}}{n} \\ & \ + \frac{ab + bc + ca}{n} + \frac{ac + ba + cb}{b} = n \end{matrix}$$

✔ (a) weighted AM–GM

✔ (b) weighted AM–GM

✔ (c) weighted AM–GM

✔ (d) sum of the three AM–GM bounds $= n$