5
Part of 2001 Tournament Of Towns
Problems(8)
Chessboard Problem
Source: ToT - 2001 Spring Junior O-Level #5
8/17/2011
On a square board divided into little squares there are rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
combinatorics unsolvedcombinatoricsToTTournament of Towns
Pawns on a Chessboard
Source: ToT - 2001 Spring Junior A-Level #5
8/17/2011
(a) One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once?
(b) Same question, but you don’t have to move the pawns in turn.
combinatorics unsolvedcombinatorics
Points on a Tetrahedron
Source: ToT - 2001 Spring Senior O-Level #5
8/17/2011
Nine points are drawn on the surface of a regular tetrahedron with an edge of cm. Prove that among these points there are two located at a distance (in space) no greater than cm.
geometry3D geometrytetrahedroncircumcirclegeometry unsolved
Rooks on a Chessboard
Source: ToT - 2001 Fall Junior A-Level #5
8/17/2011
Alex places a rook on any square of an empty chessboard. Then he places additional rooks one rook at a time, each attacking an odd number of rooks which are already on the board. A rook attacks to the left, to the right, above and below, and only the first rook in each direction. What is the maximum number of rooks Alex can place on the chessboard?
combinatorics unsolvedcombinatorics
Chess Tournament
Source: ToT - 2001 Spring Senior A-Level #5
8/17/2011
In a chess tournament, every participant played with each other exactly once, receiving point for a win, for a draw and for a loss.
(a) Is it possible that for every player , the sum of points of the players who were beaten by P is greater than the sum of points of the players who beat ?
(b) Is it possible that for every player , the first sum is less than the second one?
combinatorics unsolvedcombinatorics
Points on a Plane
Source: ToT - 2001 Fall Junior O-Level #5
8/17/2011
On the plane is a set of at least four points. If any one point from this set is removed, the resulting set has an axis of symmetry. Is it necessarily true that the whole set has an axis of symmetry?
symmetryratiogeometry unsolvedgeometry
Three Rooks
Source: ToT - 2001 Fall Senior O-Level #5
8/17/2011
The only pieces on an chessboard are three rooks. Each moves along a row or a column without running to or jumping over another rook. The white rook starts at the bottom left corner, the black rook starts in the square directly above the white rook, and the red rook starts in the square directly to the right of the white rook. The white rook is to finish at the top right corner, the black rook in the square directly to the left of the white rook, and the red rook in the square directly below the white rook. At all times, each rook must be either in the same row or the same column as another rook. Is it possible to get the rooks to their destinations?
invariantcombinatorics unsolvedcombinatorics
Prove d is a Power of 10
Source: ToT - 2001 Fall Senior A-Level #5
8/17/2011
Let and be positive integers. For any positive integer , the number contains a block of consecutive digits which constitute the number . Prove that is a power of 10.
number theory unsolvednumber theory