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OlympiadBench / 1614 / Let n be an integer greater than 1 and let X be an n-element set. A…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Expression

is multiple answer

false

language

English

question

Let

n

be an integer greater than 1 and let

X

be an

n

-element set. A non-empty collection of subsets

A_{1}, \ldots, A_{k}

X

is tight if the union

A_{1} \cup \cdots \cup A_{k}

is a proper subset of

X

and no element of

X

lies in exactly one of the

A_{i}

s. Find the largest cardinality of a collection of proper non-empty subsets of

X

, no non-empty subcollection of which is tight. Note. A subset

A

X

is proper if

A \neq X

. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

Plain-text mathematical notation (without MathML)

Let n be an integer greater than 1 and let X be an n-element set. A non-empty collection of subsets A₁,…,A_(k) of X is tight if the union A₁∪⋯∪A_(k) is a proper subset of X and no element of X lies in exactly one of the A_(i) s. Find the largest cardinality of a collection of proper non-empty subsets of X, no non-empty subcollection of which is tight.



Note. A subset A of X is proper if A≠X. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

Original LaTeX notation

Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \ldots, A_{k}$ of $X$ is tight if the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight.



Note. A subset $A$ of $X$ is proper if $A \neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1614
task_id: 78a8c16c-dea1-552d-af43-e4c2ba55b5b7
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1614
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

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