MathDB

2

Part of 2000 Cono Sur Olympiad

Problems(2)

Maximum number of 2x2 tiles on a chessboard with 1,2,...,64

Source: XI Olimpíada Matemática del Cono Sur (2000)

7/26/2011
The numbers 1,2,…,641,2,\ldots,64 are written in the squares of an 8×88\times 8 chessboard, one number to each square. Then 2×22\times 2 tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than 100100. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers 1,2,…,641,2,\ldots,64 into the squares of the chessboard that admits this maximum number of tiles.
pigeonhole principlecombinatorics unsolvedcombinatorics
Transforming a right triangle by 90 degree rotations

Source: XI Olimpíada Matemática del Cono Sur (2000)

7/26/2011
Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane 90∘90^\circ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices (0,0)(0,0), (1,0)(1,0) and (0,1)(0,1) to the triangle with vertices (0,0)(0,0), (1,0)(1,0) and (1,1)(1,1)?
geometrygeometric transformationrotationanalytic geometryinvariantgeometry unsolved