1

Part of 2007 Cono Sur Olympiad

Problems(2)

Diophantine equation x^3y + x + y = xy + 2xy^2

Source: XVIII Cono Sur Mathematical Olympiad (2007)

(x,y)

of nonnegative integers that satisfy

x^3y+x+y=xy+2xy^2.

number theory unsolvednumber theory

Almost completely coloring a chessboard

Source: XVIII Cono Sur Mathematical Olympiad (2007)

Some cells of a

2007\times 2007

table are colored. The table is charrua if none of the rows and none of the columns are completely colored.(a) What is the maximum number

k

of colored cells that a charrua table can have? (b) For such

k

, calculate the number of distinct charrua tables that exist.