Larry is the President of Logitopia

What was the puzzle?

First thing to note that it's not mentioned any where that three people are three different types; there may be 2 who are the same type.

Let's have a look at the statements made by 3.

Larry makes these two statements:

1. "The president is of a different race from the other two."
2. "Matt is not the president."

Matt makes these two statements:

1. "The president is a type B person."
2. "Larry is not the president."

Nick makes these two statements:

1. "Exactly two of us are of the same race."
2. "I am not the president."
  
We'll refer Larry's statements as L1 & L2, Matt's as M1 & M2 and those of Nick's as N1 & N2.

ANALYSIS : 

1] Suppose Matt's first statement M1 is TRUE. So, he is not a TYPE B person for sure & hence not the president. Therefore, L2 must be TRUE making sure Larry is not TYPE B person nor a president. So, Nick must be the president & TYPE B person whose both statements are FALSE. Since, N1 is turning out to be FALSE, neither Matt nor Larry can be TYPE B or both of them can't be of the same type i.e. TYPE A or TYPE C. 

However, since M2 turns out to be TRUE, Matt must be TYPE A person and hence leaving Larry as a TYPE C person. In that case, since L2 is TRUE, L1 must be FALSE. But in the case the president (TYPE B) is really different from the other two (TYPE A and TYPE C) making L1 TRUE. So, the assumption that M1 is TRUE goes wrong here in the case.

2] Now, let's suppose that Matt himself is the president. Then, L2 must be FALSE and M2, N2 would be TRUE. Since, M2 is TRUE, Matt (the president) can't be TYPE B & we know M1 is FALSE,  Matt must be TYPE C person & president.

Anyhow, Larry's first statement L1 can't be TRUE as in that case, he too will be TYPE C person as president Matt which is against the statement L1 itself. Since, his other statement L2 is FALSE, he must be TYPE B person.

Now, with N2 to be TRUE, if N1 is assumed to be TRUE then Nick would be TYPE A person. So, all three would belong to 3 different races which contradicts statement N1 itself. Hence, N1 must be FALSE. 

And if N1 FALSE and N2 TRUE, Nick would be TYPE C person as president Matt while Larry being TYPE B person. But this makes statement N1 TRUE which again contradicts our conclusion above. 

Therefore, Matt too can't be the president.

3] Suppose Nick is the president. That would make N2 FALSE and L2, M2 TRUE.
As concluded in STEP 1 above, we know M1 has to be FALSE. Then the President Nick can't be TYPE B person. With one of his false statement N2, Nick must be TYPE C person. Therefore, N1 must be TRUE. So, both Matt and Nick are TYPE C persons. 

That indicates president isn't of a different race than other two. That is L1 turns out to be FALSE. With L2 proved TRUE already, Larry would be TYPE C person. So all three would be TYPE C which contradicts TRUE statement N1.

4] Therefore, Larry must be the president.

That is L2, N2 must be TRUE and M2 must be FALSE. With M1 proved FALSE already in [1] above, Matt must be TYPE B person. Nick can't be TYPE B person with TRUE N2. 

If N1 is TRUE (i.e. Nick is TYPE A person) then Larry must be the other person with TYPE A (since Matt is TYPE B person) along with Nick for N1 to be TRUE. So, L1 too has to be TRUE. But, the president Larry is of the same race as that of Nick which is against L1 itself.

Hence, N1 must be FALSE and Nick must be TYPE C person. And therefore, L1 has to be TRUE making president Larry as a TYPE A person.

CONCLUSION : 

The President of Logitopia is Larry who is TYPE A person. Matt is TYPE B person and Nick is TYPE C person.