Given an array arr[] of N integers. Do the following operation n-1 times
It seems some kind of problem with mathematical solution (try to find the position with a closed math formula). However, here is an algorithmic approach that runs in $O(n \log^2 n)$.

Using segment tree / Fenwick tree, you can find the number of removed elements in a range (using prefix sums). Keep track of the offset (imagine having a pointer on the first element (imagine a cyclic arry and in each step we push it one element to the left instead of rotating right). Using binary search on the Fenwick tree, you can find the index of element having the distance you are looking for the offset element. Nite that after deleting half of the elements $k$ will be greater than the size and hence you can directly output the first unremoved element before the offset.