Problem of the Week

Select 52 integers at random (without replacement) from among the integers 1, 2, 3,..., 100, and label them in increasing order, say, a₁ < a₂ < a₃ < ××× < a₅₂. 

(a)  Is it always possible to solve an equation of the form a_r - a_s = a_t , with 1 £ t £ s £ r £ 52; that is, must one of these integers always be the difference of some pair of them? 

(b)  Clearly, such a difference could always be found if, instead of choosing 52 integers, you were to select all 100 of them, and, just as clearly, could be violated if you were to select a very small number of integers, say, only 2 of them.  What is the smallest number of integers you can select which will always yield a solution to such an equation?

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ã2003 Alberto L. Delgado