Question 10
(HUN 2) Three persons $A,B,C$ are playing the following game: A $k$-element subset of the set $\{1,\ldots,1986\}$ 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.)