3

Part of 2000 Cono Sur Olympiad

Problems(2)

Dividing a square into black and white rectangles

Source: XI Olimpíada Matemática del Cono Sur (2000)

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

geometryrectanglerotationgeometry unsolved

Integer divisible by the very large product of its digits

Source: XI Olimpíada Matemática del Cono Sur (2000)

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

10^{2000}

number theory unsolvednumber theory