P4

Part of 2023 Romania Team Selection Test

Problems(2)

The legendary "Sudoku" problem

Source: Romania TST 2023 Day 1 P4

4\times 4

array of pairwise distinct positive integers such that on each column, respectively row, one of the numbers is equal to the sum of the other three. Determine the least possible value of the largest number such an array may contain.
The Problem Selection Committee

Subsets of points lying in disks

Source: Romania TST 2023 Day 2 P4

Fix a positive integer

n.{}

n{}

S{}

in the plane. An eligible set is a non-empty set of the form

S\cap D,{}

D

is a closed disk in the plane. In terms of

n,

determine the smallest possible number of eligible subsets

S{}

may contain.
Proposed by Cristi Săvescu

geometrycombinatoricscombinatorial geometry