np-complete


Given a graph with n vertices and m edges, does it contain a simple cycle of length ⌈n/2⌉?


I need to prove that the following is NP-complete:
Given a graph with n vertices and m edges, does it contain a simple
cycle of length ⌈n/2⌉?
I'm familiar with the Longest Cycle problem, and I know that a Hamilton Cycle can be reduced to the Longest Cycle problem, but I'm having trouble convincing myself that this is an instance of the Longest Cycle problem.
The Longest Cycle problem states that
Given a graph and integer k, is there a cycle with no repeated nodes of at least k?
My intuition says that I can simply replace k with ⌈n/2⌉ and prove it the same way as Hamilton Cycle ≤ Longest Cycle. But that proof involves "at least k" whereas my problem deal with cycles of a specific value ⌈n/2⌉.
Any help would be appreciated.

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