MathDB

2

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

1,2,\ldots,64

are written in the squares of an

8\times 8

chessboard, one number to each square. Then

2\times 2

tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than

100

. 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,\ldots,64

into the squares of the chessboard that admits this maximum number of tiles.

Transforming a right triangle by 90 degree rotations

Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane

90^\circ

counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices

(0,0)

,

(1,0)

and

(0,1)

to the triangle with vertices

(0,0)

,

(1,0)

and

(1,1)

?

3

2

Dividing a square into black and white rectangles

Inside a

2\times 2

square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a

1\times 2

rectangle.

Integer divisible by the very large product of its digits

Is there a positive integer divisible by the product of its digits such that this product is greater than

10^{2000}

?

1

2

Line through vertices of two squares tangent to circumcircle

In square

ABCD

(labeled clockwise), let

P

be any point on

BC

and construct square

APRS

(labeled clockwise). Prove that line

CR

is tangent to the circumcircle of triangle

ABC

.

Can the digits of 16^n be nonincreasing

Call a positive integer descending if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example,

4221

and

751

are descending while

476

and

455

are not descending. Determine whether there exists a positive integer

n

for which

16^n

is descending.

2000 Cono Sur Olympiad

Subcontests

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

Transforming a right triangle by 90 degree rotations

Dividing a square into black and white rectangles

Integer divisible by the very large product of its digits

Line through vertices of two squares tangent to circumcircle

Can the digits of 16^n be nonincreasing