IMO Shortlist 1986 problem 10


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2. travnja 2012.
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Three persons A,B,C, are playing the following game:

A k-element subset of the set \{1, . . . , 1986\} is randomly chosen, with an equal probability of each choice, where k is a fixed positive integer less than or equal to 1986. The winner is A,B or C, respectively, if the sum of the chosen numbers leaves a remainder of 0, 1, or 2 when divided by 3.

For what values of k is this game a fair one? (A game is fair if the three outcomes are equally probable.)
Izvor: Međunarodna matematička olimpijada, shortlist 1986