Who's telling the truth?

In this puzzle we have $32$ people. Some of these always tell the truth and the rest always lie. Our task here is to find out the liar/s. For these, every person writes down each of these sentences:

Exactly one of these $32$ people is a liar

Exactly two of these $32$ people are liars

Exactly thirty two of these $32$ people are liars

Who is/are the liar/s? Why?

First, it is not possible that two (or more) people are telling the truth, because in that case we have that exactly $i$ and $j$ people are liars, which is not possible. Second, is not possible that everyone is a liar, because in that case everybody is telling the truth and then there are $i$ liars which is not possible. Finally, the only person telling the truth is the one saying "Exactly thirty one of these $32$ people are lying", which are the rest of the people.
Manuel Cáceres
Manuel Cáceres
Postdoctoral Researcher

My research interests include algorithmic bioinformatics, graph algorithms, string algorithms, algorithmic bioinformatics, compressed data structures, safe & complete algorithms and parameterized algorithms.