(copy-paste of my answer to your re-post; please do not ask 2 times the same question)
First of all, as others pointed out, you could state the question as such:
Statement: $\exists A,B,C. \left(A \cap B = B \cap C = A \cap C = \emptyset \right) \wedge A \cap B \cap C \neq \emptyset$
You can now either prove that this statement is false or find an example (sets A, B and C) that would satisfy it. I can't think of any such sets so I carried on with a proof:
By the definition of intersection, you know that
$$(A \cap B) \subset A \wedge (A \cap B) \subset B$$
Now if $A \cap B = \emptyset$, then
$$A \cap B \cap C = \emptyset \cap C$$
$\emptyset \cap C \subset \emptyset \Rightarrow \emptyset \cap C = \emptyset \Rightarrow A \cap B \cap C = \emptyset$
so the original statement is false.
Note that it makes intuitive sense; if the three sets do not intersect each-others in pairs, how could they possibly all intersect?
PS: drawing a Venn diagram comes in handy in this kind of question ;)