MathDB

2

Part of 2007 Middle European Mathematical Olympiad

Problems(2)

(Sub-)Sets of balls labelled with numbers

Source: MEMO Individual Competition, Question 2

9/24/2007
A set of balls contains n n balls which are labeled with numbers 1,2,3,…,n. 1,2,3,\ldots,n. We are given k>1 k > 1 such sets. We want to colour the balls with two colours, black and white in such a way, that (a) the balls labeled with the same number are of the same colour, (b) any subset of k\plus{}1 balls with (not necessarily different) labels a_{1},a_{2},\ldots,a_{k\plus{}1} satisfying the condition a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}, contains at least one ball of each colour. Find, depending on k k the greatest possible number n n which admits such a colouring.
combinatorics proposedcombinatorics
Set of 5 Points and Numbers of Acute Triangles

Source: MEMO Team Competition, Question 6

9/24/2007
For a set P P of five points in the plane, no three of them being collinear, let s(P) s(P) be the numbers of acute triangles formed by vertices in P P. Find the maximum value of s(P) s(P) over all such sets P P.
combinatorics proposedcombinatorics