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Source: RMM 2024 Problem 3

February 29, 2024

geometrycombinatoricslinear algebracombinatorial geometryRMMSets

Problem Statement

Given a positive integer

n

, a collection

\mathcal{S}

of

n-2

unordered triples of integers in

\{1,2,\ldots,n\}

is

n

-admissible if for each

1 \leq k \leq n - 2

and each choice of

k

distinct

A_1, A_2, \ldots, A_k \in \mathcal{S}

we have

\left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.

Is it true that for all

n > 3

and for each

n

-admissible collection

\mathcal{S}

, there exist pairwise distinct points

P_1, \ldots , P_n

in the plane such that the angles of the triangle

P_iP_jP_k

are all less than

61^{\circ}

for any triple

\{i, j, k\}

in

\mathcal{S}

?
Ivan Frolov, Russia

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