Please, I really need correct answer!!!Problem $5$ In this problem, we consider binomial heaps taught in class, but assume that only the

insertion operation is executed. The starting configuration is an empty heap. $($Recall that the insertion

operation is implemented by union with a single node.$)$

$(10\%)$ Prove that the amortized cost of an insertion operation is Omega$($log n$),$ where n is the total

number of nodes after all insertion operations executed.

$(10\%)$ Suppose that merging two trees has cost $1$ and all other operations has cost $0.$ Prove that

the amortized cost of an insertion operation is O$(1).$

$(5\%)$ Using the above two results, propose a new method to implement the insertion operation

for binomial heaps with O$(1)$ amortized cost. Your method must produce exactly the same heap

structure as the binomial heaps taught in class. Briefly justify your answer.