A lower bound on the time complexity of an integer problem on the uniqueness of elements

We consider an integer problem on the uniqueness of elements which we formulate as follows: Check whether there exists among $n$ given integers at least one pair equal to each other. We prove that $\Omega (n\log n)$ is the lower bound on the time complexity of this problem in a model of a linear solution tree. We obtain lower bounds on the time complexity of several geometric problems to which it reduces.