Correct solutions were received from Jimmy Chng Gim Hong, Sergiu Stefanov, Ahron Teitelman, Philippe Fondanaiche, Steve Prowse, Nancy Schwarzkopf, Jens Vob, Juan Marivela, Agustin Murillo. 

Let  A = {a₁, a₂ ,..., a₅₂} and B = {a₅₂ - a₅₁, a₅₂ - a₅₀ ,..., a₅₂ - a₁}.  The elements of A and B all lie between 1 and 100.  Since A contains 52 elements and B contains 51 elements, A and B must have an element in common.  Therefore a_r - a_s = a_t has a solution (in fact with s = r + 1).  

Clearly the same argument holds if you choose only 51 elements between 1 and 100 -- the corresponding sets A and B will have 51 and 50 elements, respectively.  Fewer than this will not guarantee a solution, since you may then select the 50 odd numbers between 1 and 100, the difference of any two of which is even.  

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