4

Part of 2010 Moldova Team Selection Test

Problems(2)

Dividing a cube

Source: Moldova TST 2010, day 1, problem 4

n\geq6

be a even natural number. Prove that any cube can be divided in \dfrac{3n(n\minus{}2)}4\plus{}2 cubes.

geometry3D geometrycombinatorics proposedcombinatorics

Hardest in ARO 2008

Source: ARO 2008, Problem 11.8

In a chess tournament 2n\plus{}3 players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least

n

next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

inductionsymmetrymodular arithmeticcombinatorial geometrycombinatorics unsolvedcombinatorics