Given 3 disjoint Turing-recognizable languages prove that one of them is decidable
Let's call MA, MB and MC the Turing machines that recognize A, B and C, respectively. The 3-TM T (a Turing machine with 3 tapes) that decides A can be described as follows:

T: input x. 


Copy x on 2nd and 3rd tapes.
T executes one step of MA on tape 1. If MA accepts x, T accepts x.
T executes one step of MB on tape 2. If MB accepts x, T rejects x.
T executes one step of MC on tape 3. If MC accepts x, T rejects x.


T halts for every string x in A U B U C, so T halts for every input in Σ*. Moreover, L(T) = A. So T is a TM that decides A, and A is Turing-decidable.