2003 Nordic

Part of Nordic

Subcontests

(4)

1

f(x) + f(y) = f(xy f(x + y))

{R^* = R-\{0\}}

be the set of non-zero real numbers. Find all functions

{f : R^* \rightarrow R^*}

{f(x) + f(y) = f(xy f(x + y))}

{x, y \in R^*}

{ x + y\ne 0 }

1

right triangle out of an interior point of an equilateral

{D}

inside the equilateral triangle

{\triangle ABC}

{\angle ADC = 150^o}

. Prove that a triangle with side lengths

{|AD|, |BD|, |CD|}

is necessarily a right-angled triangle.

1

x^3 + y^3 + z^3 − 3xyz = 2003

Find all triples of integers

{(x, y, z)}

{x^3 + y^3 + z^3 - 3xyz = 2003}

1

nordic 2003

The squares of a rectangular chessboard with 10 rows and 14 columns are colored alternatingly black and white in the usual manner. Some stones are placed the board (possibly more than one on the same square) so that there are an odd number of stones in each row and each column. Show that the total number of stones on black squares is even.