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Let's say that the statement "Only one of these two statement is true" is written on Box A, and the statement "The application is not in this box" is written on box B.

Suppose the statement on Box A is true, namely only one of the two statements is true, then the statement on Box B, "The application form is not in this box," must be false, and the application is in Box B.  

Suppose the statement on Box A is false, namely, not "only one of these two statements is true," then there are two possibilities: both of the statements on Box A and Box B are false, or, both of the two statements on Box A and Box B are true.  The latter possibility is actually impossible, because here we are supposing that the statement on Box A is false, and we cannot turn around and say it is true, too.  Consequently, the only possibility here is that both of the two statements are false.  In that case, the statement "The application form is not in this box" is false, and the application form is in Box B.

Since regardless whether the statement "Only one of the two statements is true" is factual or not, the form is in Box B, that's the box that we should open.

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