ILL 1986 Problem 43

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.)