3

Part of 1999 China National Olympiad

Problems(2)

Maximum number of connected groups among 99 space stations

Source: Chinese MO 1999

99

space stations. Each pair of space stations is connected by a tunnel. There are

99

two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of

4

space stations is called connected if one can reach each station in the group from every other station in the group without using any tunnels other than the

6

tunnels which connect them. Determine the maximum number of connected groups.

combinatorics proposedcombinatorics

Interesting colourings of a cube

Source: Chinese MO 1999

4\times4\times4

cube is composed of

64

unit cubes. The faces of

16

unit cubes are to be coloured red. A colouring is called interesting if there is exactly

1

red unit cube in every

1\times1\times 4

rectangular box composed of

4

unit cubes. Determine the number of interesting colourings.

geometry3D geometrycombinatorics proposedcombinatorics