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Omni-MATH / Three persons A,B,C, are playing the following game:

Problem

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problem

Three persons A,B,CA,B,C, are playing the following game: A kk-element subset of the set {1,...,1986}\{1, . . . , 1986\} is randomly chosen, with an equal probability of each choice, where kk is a fixed positive integer less than or equal to 19861986. The winner is A,BA,B or CC, respectively, if the sum of the chosen numbers leaves a remainder of 0,10, 1, or 22 when divided by 33. For what values of kk is this game a fair one? (A game is fair if the three outcomes are equally probable.)
Plain-text mathematical notation (without MathML)
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.)
Original LaTeX notation
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.)

Discussion

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Source and history

task_id
df86f7de-8cb5-556b-951b-ffbfcfd9d53f
task_key
test--df86f7de-8cb5-556b-951b-ffbfcfd9d53f
task_revision_id
1
config / split
/ test

Official source

initial import