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Omni-MATH / A social club has 2k+1 members, each of whom is fluent in the same k languages. Any pair of members always talk to each other in only one language. Suppose that there were no three…

Problem

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problem

A social club has

2k+1

members, each of whom is fluent in the same

k

languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let

A

be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of

A

Plain-text mathematical notation (without MathML)

A social club has 2k+1 members, each of whom is fluent in the same k languages.  Any pair of members always talk to each other in only one language.  Suppose that there were no three members such that they use only one language among them.  Let A be the number of three-member subsets such that the three distinct pairs among them use different languages.  Find the maximum possible value of A.

Original LaTeX notation

A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages.  Any pair of members always talk to each other in only one language.  Suppose that there were no three members such that they use only one language among them.  Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages.  Find the maximum possible value of $A$.

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Source and history

task_id: 28ddc7bf-2103-5e64-9121-de2dfabf7e0d
task_key: test--28ddc7bf-2103-5e64-9121-de2dfabf7e0d
task_revision_id: 2
config / split: / test

Official source

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