4

Part of 2013 Romania Team Selection Test

Problems(2)

Infinite number of sets with an intersection property

Source: Romania TST 2013 Test 2 Problem 4

k

be a positive integer larger than

1

. Build an infinite set

\mathcal{A}

\mathbb{N}

having the following properties:
(a) any

k

distinct sets of

\mathcal{A}

have exactly one common element; (b) any

k+1

distinct sets of

\mathcal{A}

have void intersection.

analytic geometryalgebrapolynomialcombinatorics proposedcombinatorics

Partition a (4n-1)-gon

Source: Romania TST 1 P4, 2013

n

be an integer greater than 1. The set

S

of all diagonals of a

\left( 4n-1\right)

-gon is partitioned into

k

S_{1},S_{2},\ldots ,S_{k},

so that, for every pair of distinct indices

i

j,

some diagonal in

S_{i}

crosses some diagonal in

S_{j};

that is, the two diagonals share an interior point. Determine the largest possible value of

k

n.

combinatorics unsolvedcombinatorics