Problem with $AM>GM>HM$ inequality

Question

Question: Let $a=1.2.3.4.5$ Then, prove that

A) $5^5>a$

B) $5^5>6a$

C) $3^5>5!$

My Approach: By the $AM>GM>HM$ inequality we have,$$\frac{1+2+3+4+5}5>a^\frac{1}{5}$$ From here we get that $$3^5>a$$, which implies $$3^5.5!$$ My question is, how do I use this approach to obtain proofs for the remaining parts?

Answer 1

Note that $$5>1,~~5>2,~~5>3,~~5>4,~~5=5$$ Multiplying these we prove that $$5^5 > 5!$$

Next note that $$4 >3,~~5>4,~~5>3,~~5>4,~~5=5$$ Multipluing them, we get $$5^5 > 6~~ 5!$$