2005 Olympic Revenge

Part of Olympic Revenge

Subcontests

(6)

1

Find the function!

Find all functions

f: R \rightarrow R

f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy

x,y \in R

1

Beautiful symmetric matrices

Let A be a symmetric matrix such that the sum of elements of any row is zero. Show that all elements in the main diagonal of cofator matrix of A are equal.

1

A geometric or a combinatoric problem?

Find all sets X of points in a plane, not all collinear, such that: For any two distinct circumferences, each contains three points of X, its intersection points are points of X.

1

The Millenium Game

Zé Roberto and Humberto are playing the Millenium Game! There are 30 empty boxes in a queue, and each box have a capacity of one blue stome. Each player takes a blue stone and places it in a box (and it is a move). The winner is who, in its move, obtain three full consecutive boxes. If Zé Roberto is the first player, who has the winner strategy?

1

A problem in counting permutations

S=\{1,2,3,\ldots,n\}

n

an odd number. Find the parity of number of permutations

\sigma : S \Rightarrow S

such that the sequence defined by

a(i)=|\sigma(i)-i|

1

A "cyclic" (maybe projective) problem

\Gamma

be a circumference, and

A,B,C,D

\Gamma

(in this order).

r

is the tangent to

\Gamma

s

is the tangent to

\Gamma

at point D. Let

E=r \cap BC,F=s \cap BC

X=r \cap s,Y=AF \cap DE,Z=AB \cap CD

Show that the points

X,Y,Z

are collinear. Note: assume the existence of all above points.