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Omni-MATH / We say a triple of real numbers (a₁,a₂,a₃) is [b]better[/b] than another triple (b₁,b₂,b₃) when exactly two out of the three following inequalities hold: a₁>b₁, a₂>b₂, a₃>b₃. We ca…

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We say a triple of real numbers (a1,a2,a3) (a_1,a_2,a_3) is [b]better[/b] than another triple (b1,b2,b3) (b_1,b_2,b_3) when exactly two out of the three following inequalities hold: a1>b1 a_1 > b_1, a2>b2 a_2 > b_2, a3>b3 a_3 > b_3. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is 1 1. For which natural numbers n n does there exist a collection S S of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of S S?
Plain-text mathematical notation (without MathML)
We say a triple of real numbers (a₁,a₂,a₃) is [b]better[/b] than another triple (b₁,b₂,b₃) when exactly two out of the three following inequalities hold: a₁>b₁, a₂>b₂, a₃>b₃. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is 1.

For which natural numbers n does there exist a collection S of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of S?
Original LaTeX notation
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.

For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

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task_id
6e60445a-e585-533a-b09a-48e93801f4e0
task_key
test--6e60445a-e585-533a-b09a-48e93801f4e0
task_revision_id
1
config / split
/ test

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