Interactive problems are somewhat rarely seen at coding contests because they involve additional effort from the problem writers. But usually they are very creative. I've made a list below. Hope you have fun solving them!

#(CLRS, agora scholarships 2004) You are given two arrays A and B of n and m elements. The elements in each array are sorted and distinct. Find the kth element in the union of two arrays using as few comparisons as possible.
#(CLRS, interview question) You are given a matrix A with n rows and m columns. The elements in each row and each column are distinct and sorted. One query is what’s the value of A[i][j]. Find out if element x is in the matrix using as few queries as possible.
#(CLRS, romanian national olympiad) Given an array A of n integers. Find the 2nd minimum in the array using as few comparisons as possible.