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Omni-MATH / Let n be an integer greater than 1. For a positive integer m, let S_(m)={1,2,…,mn}. Suppose that there exists a 2n-element set T such that (a) each element of T is an m-element sub…

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problem

Let nn be an integer greater than 11. For a positive integer mm, let Sm={1,2,…,mn}S_{m}= \{ 1,2,\ldots, mn\}. Suppose that there exists a 2n2n-element set TT such that (a) each element of TT is an mm-element subset of SmS_{m}; (b) each pair of elements of TT shares at most one common element; and (c) each element of SmS_{m} is contained in exactly two elements of TT. Determine the maximum possible value of mm in terms of nn.
Plain-text mathematical notation (without MathML)
Let n be an integer greater than 1. For a positive integer m, let S_(m)={1,2,…,mn}. Suppose that there exists a 2n-element set T such that
(a) each element of T is an m-element subset of S_(m);
(b) each pair of elements of T shares at most one common element;
and
(c) each element of S_(m) is contained in exactly two elements of T.

Determine the maximum possible value of m in terms of n.
Original LaTeX notation
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.

Determine the maximum possible value of $m$ in terms of $n$.

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Source and history

task_id
36fd56bb-eccf-58c1-af58-bebb73356c9e
task_key
test--36fd56bb-eccf-58c1-af58-bebb73356c9e
task_revision_id
1
config / split
/ test

Official source

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