No correct solutions were received from Bradley University
students.
Correct solutions were submitted by Jan Siwanowicz, Martin Dimitrov, Burkart
Venzke, Aaron Kahn, Philippe Fondanaiche, Abbas Mehrabian, Craig Macdonald,
Steven Young, Lou Cairoli.
Label the elements of the set a1,a2,..,a139
and let Sk be the sum of the first k of these
numbers, for k = 1,...,139, that is, Sk = a1
+
a2 + ... +ak , and put
S0
= 0. Among the 140 numbers S0,S1,...,
S139 there must be (at least) two which have the same
remainder upon division by 139, say Si and Sj.
But then Sj -
Si = ai+1
+ ai+2 + ... + aj
must be divisible by 139.
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