3

Part of 2004 Bulgaria Team Selection Test

Problems(4)

Table with numbers

Source: Bulgarian IMO TST 2004, Day 2, Problem 3

In any cell of an

n \times n

table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

Ross Mathematics Programinductionlinear algebramatrixcombinatorics proposedcombinatorics

Maximum value of the inradius

Source: Bulgarian IMO TST 2004, Day 1, Problem 3

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

geometryinradiusgeometry proposed

Irrational numbers

Source: Bulgarian IMO TST 2004, Day 3, Problem 3

Prove that among any

2n+1

irrational numbers there are

n+1

numbers such that the sum of any

k

of them is irrational, for all

k \in \{1,2,3,\ldots, n+1 \}

algebra proposedalgebranumber theory

A game with black and white pieces

Source: Bulgarian IMO TST 2004, Day 4, Problem 3

m

n

columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

combinatorics proposedcombinatorics