Question
If a,b,c \in R^+, prove a^{2a}b^{2b}c^{2c}\geq a^{b+c}b^{a+c}c^{a+b}
Answer
Let a\geq b\geq c
\dfrac{a^{2a}b^{2b}c^{2c}}{a^{b+c}b^{a+c}c^{a+b}} = \Big( \dfrac{a}{b} \Big) ^{a-b}\Big( \dfrac{b}{c} \Big) ^{b-c}\Big( \dfrac{a}{c} \Big) ^{a-c}\geq 1
Therefore, a^{2a}b^{2b}c^{2c}\geq a^{b+c}b^{a+c}c^{a+b}