2

Part of 2008 Middle European Mathematical Olympiad

Problems(2)

Put 2n - 2 identical pebbles on the checkerboard

Source: MEMO 2008, Single, Problem 2

n \times n

checkerboard with

n > 1, n \in \mathbb{N}.

How many possibilities are there to put 2n \minus{} 2 identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the

n \times n

analytic geometrycombinatorics unsolvedcombinatorics

Replace both of them by their sum

Source: MEMO 2008, Team, Problem 6

On a blackboard there are n \geq 2, n \in \mathbb{Z}^{\plus{}} numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers

n

for which it is possible to yield

n

identical number after a finite number of steps.

algorithmfunctioncombinatorics unsolvedcombinatorics